Try It Exploration A Exploration B Open Exploration. Fitting Integrands to Basic Rules. A Comparison of Three Similar Integrals

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1 58 CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals Section 8 Basic Integration Rules Review procedures for fitting an integrand to one of the basic integration rules Fitting Integrands to Basic Rules In this chapter, ou will stud several integration techniques that greatl epand the set of integrals to which the basic integration rules can be applied These rules are reviewed on page 5 A major step in solving an integration problem is recognizing which basic integration rule to use As shown in Eample, slight differences in the integrand can lead to ver different solution techniques EXAMPLE A Comparison of Three Similar Integrals EXPLORATION A Comparison of Three Similar Integrals Which, if an, of the following integrals can be evaluated using the basic integration rules? For an that can be evaluated, do so For an that can t, eplain wh a d b d c d Find each integral a b c 9 d 9 d Solution a Use the Arctangent Rule and let u and a 9 d d arctan C arctan C Constant Multiple Rule Arctangent Rule Simplif b Here the Arctangent Rule does not appl because the numerator contains a factor of Consider the Log Rule and let u 9 Then du d, and ou have Constant Multiple Rule 9 d d 9 9 d du u Substitution: u 9 NOTE Notice in Eample (c) that some preinar algebra is required before appling the rules for integration, and that subsequentl more than one rule is needed to evaluate the resulting integral Log Rule c Because the degree of the numerator is equal to the degree of the denominator, ou should first use division to rewrite the improper rational function as the sum of a polnomial and a proper rational function ln u C ln 9 C 9 d 6 9 d d 6 9 d 6 arctan C arctan C Rewrite using long division Write as two integrals Integrate Simplif Tr It Eploration A Eploration B Open Eploration

2 SECTION 8 Basic Integration Rules 59 = + EXAMPLE Evaluate Using Two Basic Rules to Solve a Single Integral d Solution Begin b writing the integral as the sum of two integrals Then appl the Power Rule and the Arcsine Rule as follows d d d d d The area of the region is approimatel 89 Figure 8 See Figure 8 arcsin 89 Editable Graph Tr It Eploration A Eploration B TECHNOLOGY Simpson s Rule can be used to give a good approimation of the value of the integral in Eample (for n, the approimation is 89) When using numerical integration, however, ou should be aware that Simpson s Rule does not alwas give good approimations when one or both of the its of integration are near a vertical asmptote For instance, using the Fundamental Theorem of Calculus, ou can obtain EXAMPLE Find 99 d 6 Appling Simpson s Rule (with n ) to this integral produces an approimation of d A Substitution Involving a u STUDY TIP Rules 8, 9, and of the basic integration rules on the net page all have epressions involving the sum or difference of two squares: a u a u u a With such an epression, consider the substitution as in Eample u f, Solution Because the radical in the denominator can be written in the form a u ou can tr the substitution u Then du d, and ou have 6 d 6 arcsin u C d 6 du u arcsin C Rewrite integral Substitution: u Arcsine Rule Rewrite as a function of Tr It Eploration A Eploration B

3 5 CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals Surprisingl, two of the most commonl overlooked integration rules are the Log Rule and the Power Rule Notice in the net two eamples how these two integration rules can be disguised EXAMPLE Find e d A Disguised Form of the Log Rule Review of Basic Integration Rules a > kfu du kf u du f u ± gu du du u C 5 6 e u du e u C 7 f u du ±gu du u n du un C, n du u ln u C a u du ln a au C 8 sin u du cos u C n Solution The integral does not appear to fit an of the basic rules However, the quotient form suggests the Log Rule If ou let u e, then du e d You can obtain the required du b adding and subtracting e in the numerator, as follows e d e e e d Tr It Add and subtract Rewrite as two fractions Rewrite as two integrals Integrate in numerator NOTE There is usuall more than one wa to solve an integration problem For instance, in Eample, tr integrating b multipling the numerator and denominator b e to obtain an integral of the form duu See if ou can get the same answer b this procedure (Be careful: the answer will appear in a different form) EXAMPLE 5 e e e e d e d e ln e C Eploration A d A Disguised Form of the Power Rule e 9 cos u du sin u C Find cot lnsin d tan u du ln cos u C cot u du ln sin u C csc u du sec u du tan u C 5 csc u du cot u C 6 sec u tan u du sec u C sec u du ln sec u tan u C lncsc u cot u C csc u cot u du csc u C du a u arcsin u a C du a u a arctan u a C du uu a a arcsec u a C Solution Again, the integral does not appear to fit an of the basic rules However, considering the two primar choices for u u cot and u lnsin, ou can see that the second choice is the appropriate one because So, NOTE u lnsin and cot lnsin d u du In Eample 5, tr checking that the derivative of lnsin C u C is the integrand of the original integral du cos sin d lnsin C cot d Substitution: u lnsin Integrate Tr It Eploration A Technolog Rewrite as a function of

4 SECTION 8 Basic Integration Rules 5 Trigonometric identities can often be used to fit integrals to one of the basic integration rules EXAMPLE 6 Using Trigonometric Identities Find tan d TECHNOLOGY If ou have access to a computer algebra sstem, tr using it to evaluate the integrals in this section Compare the form of the antiderivative given b the software with the form obtained b hand Sometimes the forms will be the same, but often the will differ For instance, wh is the antiderivative ln C equivalent to the antiderivative ln C? Solution Note that tan u is not in the list of basic integration rules However, sec u is in the list This suggests the trigonometric identit tan u sec u If ou let u, then du d and tan d tan u du sec u du sec u du du tan u u C Substitution: u Trigonometric identit Rewrite as two integrals Integrate tan C Rewrite as a function of Tr It Eploration A This section concludes with a summar of the common procedures for fitting integrands to the basic integration rules Procedures for Fitting Integrands to Basic Rules Technique Epand (numerator) Separate numerator Complete the square Divide improper rational function Add and subtract terms in numerator Use trigonometric identities Multipl and divide b Pthagorean conjugate Eample e e e cot csc sin sin sin sin sin cos sec sin cos sin sin NOTE Remember that ou can separate numerators but not denominators Watch out for this common error when fitting integrands to basic rules Do not separate denominators

5 5 CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals Eercises for Section 8 The smbol Click on Click on indicates an eercise in which ou are instructed to use graphing technolog or a smbolic computer algebra sstem to view the complete solution of the eercise to print an enlarged cop of the graph In Eercises, select the correct antiderivative d d (a) C (a) (c) arctan C (a) (c) arctan C (a) sin C (c) sin C In Eercises 5, select the basic integration formula ou can use to find the integral, and identif u and a when appropriate 5 d 6 t t t dt (b) (d) (b) (d) (b) (d) (b) (d) 7 8 d In Eercises 5 5, find the indefinite integral d 6 t 9 dt t t dt z dz 9 5 v d v dv (c) C d d ln C d d ln C d d cos C ln C C ln C C ln C sin C sin C 9 t dt d t sin t dt sec tan d cos esin d d t t d 9t dt d d e 5 6 e d d t dt 7 d 8 9 cos d sec d csc cot d cos d csc ecot e 5 d d e e d d ln 7 d 8 tan lncos d sin cos 9 d cos sin cos d sec d t dt d e 5 tant dt 6 t t t dt Slope Fields In Eercises 5 5, a differential equation, a point, and a slope field are given (a) Sketch two approimate solutions of the differential equation on the slope field, one of which passes through the given point (b) Use integration to find the particular solution of the differential equation and use a graphing utilit to graph the solution Compare the result with the sketches in part (a) To print an enlarged cop of the graph, select the MathGraph button ds 5 5 dt t t 6 d 8 d 65 d d, s t d d d tan, sin d

6 SECTION 8 Basic Integration Rules 5 d 5 sec tan 5 d, 9 d d, Slope Fields In Eercises 55 and 56, use a computer algebra sstem to graph the slope field for the differential equation and graph the solution through the specified initial condition In Eercises 57 6, solve the differential equation d d e 59 tan sec 6 In Eercises 6 68, evaluate the definite integral Use the integration capabilities of a graphing utilit to verif our result 6 cos d 6 6 e d d 68 9 d Area 9 d, d d 5, d In Eercises 69 7, find the area of the region dr dt et e t e sin t cos t dt ln d d 5 d 7 7 sin In Eercises 75 78, use a computer algebra sstem to find the integral Use the computer algebra sstem to graph two antiderivatives Describe the relationship between the two graphs of the antiderivatives d d sin d 78 e e d 5 Writing About Concepts In Eercises 79 8, state the integration formula ou would use to perform the integration Eplain wh ou chose that formula Do not integrate (5, ) 79 d d sec tan d d 8 Eplain wh the antiderivative e C is equivalent to the antiderivative Ce 8 Eplain wh the antiderivative sec C is equivalent to the antiderivative tan C

7 5 CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals 85 Determine the constants a and b such that Use this result to integrate 86 Area The graphs of f and g a intersect at the points, and a, a Find a a > such that the area of the region bounded b the graphs of these two functions is 87 Think About It Use a graphing utilit to graph the function f 5 Use the graph to determine whether 7 5 f d is positive or negative Eplain 88 Think About It When evaluating d is it appropriate to substitute u, u, and d to obtain Eplain Approimation In Eercises 89 and 9, determine which value best approimates the area of the region between the -ais and the function over the given interval (Make our selection on the basis of a sketch of the region and not b integrating) 89 9 sin cos a sin b (a) (b) (c) 8 (d) 8 (e) (a) (b) (c) (d) (e) Interpreting Integrals In Eercises 9 and 9, (a) sketch the region whose area is given b the integral, (b) sketch the solid whose volume is given b the integral if the disk method is used, and (c) sketch the solid whose volume is given b the integral if the shell method is used (There is more than one correct answer for each part) f u du?, f, d,, Volume The region bounded b e,,, and b b > is revolved about the -ais (a) Find the volume of the solid generated if b d sin cos (b) Find b such that the volume of the generated solid is cubic units 9 Arc Length Find the arc length of the graph of lnsin from to 95 Surface Area Find the area of the surface formed b revolving the graph of on the interval, 9 about the -ais d du u 96 Centroid Find the -coordinate of the centroid of the region bounded b the graphs of and In Eercises 97 and 98, find the average value of the function over the given interval , f, 98 f sin n, n, n is a positive integer Arc Length In Eercises 99 and, use the integration capabilities of a graphing utilit to approimate the arc length of the curve over the given interval 99 tan,,, Finding a Pattern (a) Find cos d (b) Find cos 5 d (c) Find cos 7 d (d) Eplain how to find cos 5 d integrating Finding a Pattern without actuall (a) Write tan d in terms of tan d Then find tan d (b) Write tan 5 d in terms of tan d (c) Write tan k d, where k is a positive integer, in terms of tan k d (d) Eplain how to find integrating without actuall Methods of Integration Show that the following results are equivalent Integration b, tables:, tan 5 d, 8 d ln C Integration b computer algebra sstem: d arcsinh C Evaluate Putnam Eam Challenge ln9 d ln9 ln This problem was composed b the Committee on the Putnam Prize Competition The Mathematical Association of America All rights reserved

8 SECTION 8 Integration b Parts 55 Section 8 EXPLORATION Proof Without Words Here is a different approach to proving the formula for integration b parts Eercise taken from Proof Without Words: Integration b Parts b Roger B Nelsen, Mathematics Magazine, April 99, b permission of the author s = g(b) r = g(a) v u = f() v = g() Integration b Parts Find an antiderivative using integration b parts Use a tabular method to perform integration b parts Integration b Parts In this section ou will stud an important integration technique called integration b parts This technique can be applied to a wide variet of functions and is particularl useful for integrands involving products of algebraic and transcendental functions For instance, integration b parts works well with integrals such as ln d, e d, and Integration b parts is based on the formula for the derivative of a product d dv du uv u v d d d uv vu where both u and v are differentiable functions of If and are continuous, ou can integrate both sides of this equation to obtain uv uv d vu d e sin d u v p = f(a) u q = f(b) u dv v du Area Area qs pr s r r s p u dv v du q uv q,s p,r u dv uv q,s p p,r Eplain how this graph proves the theorem Which notation in this proof is unfamiliar? What do ou think it means? q v du B rewriting this equation, ou obtain the following theorem THEOREM 8 Integration b Parts If u and v are functions of and have continuous derivatives, then u dv uv v du This formula epresses the original integral in terms of another integral Depending on the choices of u and dv, it ma be easier to evaluate the second integral than the original one Because the choices of u and dv are critical in the integration b parts process, the following guidelines are provided Guidelines for Integration b Parts Tr letting dv be the most complicated portion of the integrand that fits a basic integration rule Then u will be the remaining factor(s) of the integrand Tr letting u be the portion of the integrand whose derivative is a function simpler than u Then dv will be the remaining factor(s) of the integrand

9 56 CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals EXAMPLE Integration b Parts Find e d Solution To appl integration b parts, ou need to write the integral in the form u dv There are several was to do this e d, e d, e d, e d u dv u dv u dv u dv The guidelines on page 55 suggest choosing the first option because the derivative of u is simpler than, and dv e d is the most complicated portion of the integrand that fits a basic integration formula dv e d v dv e d e NOTE In Eample, note that it is not necessar to include a constant of integration when solving v e d e C To illustrate this, replace v e b v e C and appl integration b parts to see that ou obtain the same result u du d Now, integration b parts produces u dv uv v du e d e e d e e C Integration b parts formula Substitute Integrate To check this, differentiate e e C to see that ou obtain the original integrand Tr It Eploration A Eploration B EXAMPLE Integration b Parts Find ln d Solution In this case, is more easil integrated than ln Furthermore, the derivative of ln is simpler than ln So, ou should let dv d dv d u ln v d du d Integration b parts produces u dv uv v du Integration b parts formula TECHNOLOGY Tr graphing and ln d ln 9 on our graphing utilit Do ou get the same graph? (This will take a while, so be patient) ln d ln d ln d ln C 9 You can check this result b differentiating Substitute Simplif Integrate d d ln 9 ln ln Tr It Eploration A Eploration B

10 SECTION 8 Integration b Parts 57 One surprising application of integration b parts involves integrands consisting of a single term, such as ln d or arcsin d In these cases, tr letting dv d, as shown in the net eample EXAMPLE An Integrand with a Single Term Evaluate arcsin d Solution Let dv d dv d v d u arcsin du d Integration b parts now produces π π (, ) = arcsin u dv uv v du arcsin d arcsin d arcsin d Integration b parts formula Substitute Rewrite The area of the region is approimatel 57 Figure 8 arcsin C Integrate Using this antiderivative, ou can evaluate the definite integral as follows arcsin d arcsin 57 The area represented b this definite integral is shown in Figure 8 Editable Graph Tr It Eploration A Eploration B TECHNOLOGY Remember that there are two was to use technolog to evaluate a definite integral: () ou can use a numerical approimation such as the Trapezoidal Rule or Simpson s Rule, or () ou can use a computer algebra sstem to find the antiderivative and then appl the Fundamental Theorem of Calculus Both methods have shortcomings To find the possible error when using a numerical method, the integrand must have a second derivative (Trapezoidal Rule) or a fourth derivative (Simpson s Rule) in the interval of integration: the integrand in Eample fails to meet either of these requirements To appl the Fundamental Theorem of Calculus, the smbolic integration utilit must be able to find the antiderivative Which method would ou use to evaluate arctan d? Which method would ou use to evaluate arctan d?

11 58 CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals Some integrals require repeated use of the integration b parts formula EXAMPLE Repeated Use of Integration b Parts Find sin d Solution The factors and sin are equall eas to integrate However, the derivative of becomes simpler, whereas the derivative of sin does not So, ou should let u dv sin d v sin d cos u du d Now, integration b parts produces sin d cos cos d First use of integration b parts This first use of integration b parts has succeeded in simplifing the original integral, but the integral on the right still doesn t fit a basic integration rule To evaluate that integral, ou can appl integration b parts again This time, let u dv cos d v cos d sin u du d Now, integration b parts produces cos d sin sin d Second use of integration b parts sin cos C Combining these two results, ou can write sin d cos sin cos C Tr It Eploration A Eploration B Eploration C When making repeated applications of integration b parts, ou need to be careful not to interchange the substitutions in successive applications For instance, in Eample, the first substitution was u and dv sin d If, in the second application, ou had switched the substitution to u cos and dv, ou would have obtained sin d cos cos d Tr to find EXPLORATION e cos d b letting u cos and dv e d in the first substitution For the second substitution, let u sin and dv e d cos cos sin d sin d thereb undoing the previous integration and returning to the original integral When making repeated applications of integration b parts, ou should also watch for the appearance of a constant multiple of the original integral For instance, this occurs when ou use integration b parts to evaluate e cos d, and also occurs in the net eample

12 SECTION 8 Integration b Parts 59 NOTE The integral in Eample 5 is an important one In Section 8 (Eample 5), ou will see that it is used to find the arc length of a parabolic segment EXAMPLE 5 Integration b Parts Find sec d Solution The most complicated portion of the integrand that can be easil integrated is sec, so ou should let dv sec d and u sec dv sec d v sec d tan u sec du sec tan d Integration b parts produces sec d sec tan sec tan d Substitute u dv uv v du Integration b parts formula STUDY TIP identities sin cos The trigonometric cos cos pla an important role in this chapter sec d sec tan sec sec d sec d sec tan sec d sec d sec d sec tan sec d sec d sec tan ln sec tan C Trigonometric identit Rewrite Collect like integrals Integrate and divide b Tr It Eploration A Eploration B EXAMPLE 6 Finding a Centroid = sin π (, ) A machine part is modeled b the region bounded b the graph of sin and the -ais,, as shown in Figure 8 Find the centroid of this region Figure 8 sin Editable Graph π Solution A Begin b finding the area of the region Now, ou can find the coordinates of the centroid as follows sin d cos d sin A sin sin d cos You can evaluate the integral for, A sin d, with integration b parts To do this, let dv sin d and u This produces v cos and du d, and ou can write 8 sin d cos cos d cos sin C Finall, ou can determine to be A sin d cos sin So, the centroid of the region is, 8 Tr It Eploration A

13 5 CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals As ou gain eperience in using integration b parts, our skill in determining u and dv will increase The following summar lists several common integrals with suggestions for the choices of u and dv STUDY TIP You can use the acronm LIATE as a guideline for choosing u in integration b parts In order, check the integrand for the following Is there a Logarithmic part? Is there an Inverse trigonometric part? Is there an Algebraic part? Is there a Trigonometric part? Is there an Eponential part? Summar of Common Integrals Using Integration b Parts For integrals of the form or n e a d, n sin a d, n cos a d let u n and let dv e a d, sin a d, or cos a d For integrals of the form or n ln d, n arcsin a d, n arctan a d let u ln, arcsin a, or arctan a and let dv n d For integrals of the form ea sin b d or e a cos b d let u sin b or cos b and let dv e a d Tabular Method In problems involving repeated applications of integration b parts, a tabular method, illustrated in Eample 7, can help to organize the work This method works well for integrals of the form n sin a d, n cos a d, and n e a d EXAMPLE 7 Using the Tabular Method Find sin d FOR FURTHER INFORMATION For more information on the tabular method, see the article Tabular Integration b Parts b David Horowitz in The College Mathematics Journal, and the article More on Tabular Integration b Parts b Leonard Gillman in The College Mathematics Journal MathArticle Solution Begin as usual b letting u and dv v d sin d Net, create a table consisting of three columns, as shown Alternate Signs 6 u and Its Derivatives Differentiate until ou obtain as a derivative vand Its Antiderivatives sin sin 6 cos The solution is obtained b adding the signed products of the diagonal entries: sin d cos sin cos C 8 cos MathArticle Tr It Open Eploration

14 SECTION 8 Integration b Parts 5 Eercises for Section 8 The smbol Click on Click on indicates an eercise in which ou are instructed to use graphing technolog or a smbolic computer algebra sstem to view the complete solution of the eercise to print an enlarged cop of the graph In Eercises, match the antiderivative with the correct integral [Integrals are labeled (a), (b), (c), and (d)] (a) ln d (c) e d sin cos sin cos sin e e e ln (b) sin d (d) cos d In Eercises 5, identif u and dv for finding the integral using integration b parts (Do not evaluate the integral) 5 e d 6 e d In Eercises 7, solve the differential equation e 7 8 d 9 dt t t cos ln d d arctan Slope Fields In Eercises and, a differential equation, a point, and a slope field are given (a) Sketch two approimate solutions of the differential equation on the slope field, one of which passes through the given point (b) Use integration to find the particular solution of the differential equation and use a graphing utilit to graph the solution Compare the result with the sketches in part (a) To print an enlarged cop of the graph select the MathGraph button 7 ln d 8 9 sec d ln d cos d d cos,, d d d e sin, 5, 8 7 In Eercises 6, find the integral (Note: Solve b the simplest method not all require integration b parts) e d e d 6 e d 5 e d 6 7 t lnt dt 8 ln 9 d e d e d 5 d 6 7 cos d 8 9 sin d t csc t cot t dt arctan d 5 e sin d 6 et t dt ln d ln d ln d ln d e d d sin d cos d sec tan d arccos d e cos d Slope Fields In Eercises 5 and 6, use a computer algebra sstem to graph the slope field for the differential equation and graph the solution through the specified initial condition d d 5 6 d d e8 sin In Eercises 7 58, evaluate the definite integral Use a graphing utilit to confirm our result 7 e d 8 9 cos d 5 5 arccos d 5 5 e sin d 5 55 ln d 56 ln d 57 arcsec d 58 sec d 5 e d sin d arcsin d e cos d

15 5 CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals In Eercises 59 6, use the tabular method to find the integral 59 6 e d e d 6 6 sin d cos d 6 sec 6 d d In Eercises 65 7, find or evaluate the integral using substitution first, then using integration b parts 65 sin d 66 cos d 67 d 68 e d 69 cosln d 7 ln d Writing About Concepts 7 Integration b parts is based on what differentiation rule? Eplain 7 In our own words, state guidelines for integration b parts In Eercises 7 78, state whether ou would use integration b parts to evaluate the integral If so, identif what ou would use for u and dv Eplain our reasoning ln 7 7 ln d d 75 e d 76 e d d d In Eercises 79 8, use a computer algebra sstem to (a) find or evaluate the integral and (b) graph two antiderivatives (c) Describe the relationship between the graphs of the antiderivatives 79 t 8 e t dt sin d 8 e sin d 8 8 Integrate d (a) b parts, letting dv d (b) b substitution, letting u 8 Integrate d (a) b parts, letting dv d (b) b substitution, letting u 5 5 d 85 Integrate (a) b parts, letting dv d (b) b substitution, letting u 86 Integrate d (a) b parts, letting dv d (b) b substitution, letting u In Eercises 87 and 88, use a computer algebra sstem to find the integral for n,,, and Use the result to obtain a general rule for the integral for an positive integer n and test our results for n n ln d n e d In Eercises 89 9, use integration b parts to verif the formula (For Eercises 89 9, assume that n is a positive integer) In Eercises 95 98, find the integral b using the appropriate formula from Eercises ln d Area In Eercises 99, use a graphing utilit to graph the region bounded b the graphs of the equations, and find the area of the region 99 n sin d n cos n n cos d n cos d n sin n n sin d n ln d n e a n e d a n a a n e a d e a sin b d ea a sin b b cos b a b e a cos b d ea a cos b b sin b a b cos d e cos d e d d e,, 9 e,,, e sin,,, sin,,, n n n ln C C C

16 SECTION 8 Integration b Parts 5 Area, Volume, and Centroid Given the region bounded b the graphs of ln,, and e, find (a) the area of the region (b) the volume of the solid generated b revolving the region about the -ais (c) the volume of the solid generated b revolving the region about the -ais (d) the centroid of the region Volume and Centroid Given the region bounded b the graphs of sin,,, and, find (a) the volume of the solid generated b revolving the region about the -ais (b) the volume of the solid generated b revolving the region about the -ais (c) the centroid of the region 5 Centroid Find the centroid of the region bounded b the graphs of arcsin,, and How is this problem related to Eample 6 in this section? 6 Centroid Find the centroid of the region bounded b the graphs of f, g,, and 7 Average Displacement A damping force affects the vibration of a spring so that the displacement of the spring is given b e t cos t 5 sin t Find the average value of on the interval from t to t 8 Memor Model A model for the abilit M of a child to memorize, measured on a scale from to, is given b M 6t ln t, < t, where t is the child s age in ears Find the average value of this model (a) between the child s first and second birthdas (b) between the child s third and fourth birthdas Present Value In Eercises 9 and, find the present value P of a continuous income flow of ct dollars per ear if P t where t is the time in ears and r is the annual interest rate compounded continuousl 9 Integrals Used to Find Fourier Coefficients In Eercises and, verif the value of the definite integral, where n is a positive integer cte rt dt ct, t, r 5%, t ct, 5t, r 7%, t 5, n sin n d, n cos n d n n n is odd n is even Vibrating String A string stretched between the two points, and, is plucked b displacing the string h units a its midpoint The motion of the string is modeled b a Fourier Sine Series whose coefficients are given b b n h sin n Find b n Find the fallac in the following argument that dv d u d d d So, 5 Let f be positive and strictl increasing on the interva < a b Consider the region R bounded b the graphs of f,, a, and b If R is revolved abou the -ais, show that the disk method and shell method ield the same volume 6 Euler s Method Consider the differential equation f e with the initial condition f (a) Use integration to solve the differential equation (b) Use a graphing utilit to graph the solution of the differential equation (c) Use Euler s Method with h 5, and the recursive capabilities of a graphing utilit, to generate the first 8 points of the graph of the approimate solution Use the graphing utilit to plot the points Compare the result with the graph in part (b) (d) Repeat part (c) using h and generate the first points (e) Wh is the result in part (c) a better approimation of the solution than the result in part (d)? Euler s Method In Eercises 7 and 8, consider the differential equation and repeat parts (a) (d) of Eercise 6 7 f sin 8 f cos f d h v d du d f 9 Think About It Give a geometric eplanation to eplain wh sin d d Verif the inequalit b evaluating the integrals Finding a Pattern Find the area bounded b the graphs of sin and over each interval (a), (b), (c), sin n d Describe an patterns that ou notice What is the area between the graphs of sin and over the interva n, n, where n is an nonnegative integer? Eplain

17 5 CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals Section 8 Trigonometric Integrals Solve trigonometric integrals involving powers of sine and cosine Solve trigonometric integrals involving powers of secant and tangent Solve trigonometric integrals involving sine-cosine products with different angles SHEILA SCOTT MACINTYRE (9 96) Sheila Scott Macintre published her first paper on the asmptotic periods of integral functions in 95 She completed her doctorate work at Aberdeen Universit, where she taught In 958 she accepted a visiting research fellowship at the Universit of Cincinnati Integrals Involving Powers of Sine and Cosine In this section ou will stud techniques for evaluating integrals of the form sin m cos n d and where either m or n is a positive integer To find antiderivatives for these forms, tr to break them into combinations of trigonometric integrals to which ou can appl the Power Rule For instance, ou can evaluate sin 5 cos d with the Power Rule b letting u sin Then, du cos d and ou have sin 5 cos d u 5 du u6 6 C sin6 C 6 To break up sinm cos n d into forms to which ou can appl the Power Rule, use the following identities sin cos sin cos cos cos sec m tan n d Pthagorean identit Guidelines for Evaluating Integrals Involving Sine and Cosine Half-angle identit for sin Half-angle identit for cos If the power of the sine is odd and positive, save one sine factor and convert the remaining factors to cosines Then, epand and integrate Odd Convert to cosines Save for du sin k cos n d sin k cosn sin d cos k cos n sin d If the power of the cosine is odd and positive, save one cosine factor and convert the remaining factors to sines Then, epand and integrate Odd Convert to sines Save for du sin m cos k d sin m cos k cos d sin m sin k cos d If the powers of both the sine and cosine are even and nonnegative, make repeated use of the identities sin cos and cos cos to convert the integrand to odd powers of the cosine Then proceed as in guideline

18 SECTION 8 Trigonometric Integrals 55 TECHNOLOGY Use a computer algebra sstem to find the integral in Eample You should obtain sin cos d cos 5 7 sin 5 C Is this equivalent to the result obtained in Eample? EXAMPLE Find sin cos d Power of Sine Is Odd and Positive Solution Because ou epect to use the Power Rule with u cos, save one sine factor to form du and convert the remaining sine factors to cosines sin cos d sin cos sin d cos cos sin d Rewrite Trigonometric identit cos cos 6 sin d Multipl cos sin d cos 6 sin d Rewrite cos sin d cos 6 sin d cos5 5 cos7 7 C Tr It Eploration A Eploration B Integrate Eploration C Technolog In Eample, both of the powers m and n happened to be positive integers However, the same strateg will work as long as either m or n is odd and positive For instance, in the net eample the power of the cosine is, but the power of the sine is 8 6 π 6 = cos sin π The area of the region is approimatel 9 Figure 8 Editable Graph EXAMPLE Evaluate Power of Cosine Is Odd and Positive Solution Because ou epect to use the Power Rule with u sin, save one cosine factor to form du and convert the remaining cosine factors to sines cos sin d cos cos d sin 6 6 cos sin d sin 9 sin cos sin sin cos sin cos d sin Figure 8 shows the region whose area is represented b this integral Tr It Eploration A Open Eploration d

19 56 CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals EXAMPLE Power of Cosine Is Even and Nonnegative Find cos d Solution Because m and n are both even and nonnegative m, ou can replace cos b cos cos cos d d cos cos cos d cos d 8d cos d cos d 8 sin sin C Use a smbolic differentiation utilit to verif this Can ou simplif the derivative to obtain the original integrand? Tr It Eploration A Eploration B In Eample, if ou were to evaluate the definite integral from to, ou would obtain cos d sin sin Note that the onl term that contributes to the solution is 8 This observation is generalized in the following formulas developed b John Wallis JOHN WALLIS (66 7) Wallis did much of his work in calculus prior to Newton and Leibniz, and he influenced the thinking of both of these men Wallis is also credited with introducing the present smbol for infinit MathBio Wallis s Formulas If n is odd n, then cos n d n n If n is even n, then cos n d 5 6 n n These formulas are also valid if cos n is replaced b sin n (You are asked to prove both formulas in Eercise )

20 SECTION 8 Trigonometric Integrals 57 Integrals Involving Powers of Secant and Tangent The following guidelines can help ou evaluate integrals of the form sec m tan n d Guidelines for Evaluating Integrals Involving Secant and Tangent If the power of the secant is even and positive, save a secant-squared factor and convert the remaining factors to tangents Then epand and integrate Even Convert to tangents Save for du sec k tan n d sec k tan n sec d tan k tan n sec d If the power of the tangent is odd and positive, save a secant-tangent factor and convert the remaining factors to secants Then epand and integrate Odd Convert to secants Save for du sec m tank d secm tan k sec tan d secm sec k sec tan d If there are no secant factors and the power of the tangent is even and positive, convert a tangent-squared factor to a secant-squared factor, then epand and repeat if necessar Convert to secants tan n d tan n tan d tan n sec d If the integral is of the form sec m d, where m is odd and positive, use integration b parts, as illustrated in Eample 5 in the preceding section 5 If none of the first four guidelines applies, tr converting to sines and cosines EXAMPLE Power of Tangent Is Odd and Positive Find tan sec d Solution Because ou epect to use the Power Rule with u sec, save a factor of sec tan to form du and convert the remaining tangent factors to secants tan sec d sec tan d sec tan sec tan d sec sec sec tan d sec sec sec tan d sec sec C Tr It Eploration A Eploration B

21 58 CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals NOTE In Eample 5, the power of the tangent is odd and positive So, ou could also find the integral using the procedure described in guideline on page 57 In Eercise 85, ou are asked to show that the results obtained b these two procedures differ onl b a constant EXAMPLE 5 Power of Secant Is Even and Positive Find sec tan d Solution Let u tan, then du sec d and ou can write sec tan d sec tan sec d tan tan sec d tan tan 5 sec d tan tan tan6 6 C tan6 8 C Tr It Eploration A EXAMPLE 6 Power of Tangent Is Even Evaluate tan d Solution Because there are no secant factors, ou can begin b converting a tangentsquared factor to a secant-squared factor tan d tan tan d = tan ( π, ) tan sec d tan sec d tan d tan sec d sec d 5 π 8 π The area of the region is approimatel 9 Figure 85 tan You can evaluate the definite integral as follows tan tan d 9 tan C tan The area represented b the definite integral is shown in Figure 85 Tr using Simpson s Rule to approimate this integral With n 8, ou should obtain an approimation that is within of the actual value Editable Graph Tr It Eploration A Eploration B

22 SECTION 8 Trigonometric Integrals 59 For integrals involving powers of cotangents and cosecants, ou can follow a strateg similar to that used for powers of tangents and secants Also, when integrating trigonometric functions, remember that it sometimes helps to convert the entire integrand to powers of sines and cosines EXAMPLE 7 Converting to Sines and Cosines sec Find tan d Solution Because the first four guidelines on page 57 do not appl, tr converting the integrand to sines and cosines In this case, ou are able to integrate the resulting powers of sine and cosine as follows sec tan d cos cos sin d sin cos d Tr It sin C csc C Eploration A FOR FURTHER INFORMATION To learn more about integrals involving sine-cosine products with different angles, see the article Integrals of Products of Sine and Cosine with Different Arguments b Sherrie J Nicol in The College Mathematics Journal MathArticle Integrals Involving Sine-Cosine Products with Different Angles Integrals involving the products of sines and cosines of two different angles occur in man applications In such instances ou can use the following product-to-sum identities sin m sin n cosm n cosm n sin m cos n sinm n sinm n cos m cos n cosm n cosm n EXAMPLE 8 Using Product-to-Sum Identities Find sin 5 cos d Solution Considering the second product-to-sum identit above, ou can write sin 5 cos d sin sin 9 d cos 9 cos 9 C cos cos 9 8 C Tr It Eploration A

23 5 CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals Eercises for Section 8 The smbol Click on Click on indicates an eercise in which ou are instructed to use graphing technolog or a smbolic computer algebra sstem to view the complete solution of the eercise to print an enlarged cop of the graph In Eercises, use differentiation to match the antiderivative with the correct integral [Integrals are labeled (a), (b), (c), and (d)] (a) sin tan d (c) sin sec d sec cos sec tan tan In Eercises 5 8, find the integral (b) 8cos d (d) tan d sin cos sin cos 5 cos sin d 6 cos sin d sec tan d tan t sec t dt 5 tan sec d 6 tan 5 sec d 7 sec 6 tan d 8 9 sec tan d tan sec d In Eercises 6, solve the differential equation dr ds sin sin cos d tan sec 5 6 sec tan d tan d tan sec 5 d d tan sec 7 sin 5 cos d 8 9 sin 5 cos d cos sin d cos d 5 sin cos d 6 7 sin d 8 sin d cos d sin5 t cos t dt sin d sin d sin d Slope Fields In Eercises 7 and 8, a differential equation, a point, and a slope field are given (a) Sketch two approimate solutions of the differential equation on the slope field, one of which passes through the given point (b) Use integration to find the particular solution of the differential equation and use a graphing utilit to graph the solution Compare the result with the sketches in part (a) To print an enlarged cop of the graph select the MathGraph button d 7 8 d sin,, d d sec tan,, 5 In Eercises 9, use Wallis s Formulas to evaluate the integral 9 cos d cos 7 d sin 6 d cos 5 d In Eercises 5, find the integral involving secant and tangent 5 sec d 6 sec d sin d sin 7 d 5 Slope Fields In Eercises 9 and 5, use a computer algebra sstem to graph the slope field for the differential equation, and graph the solution through the specified initial condition d sin 9, 5 d In Eercises 5 5, find the integral 5 d d tan, 5 7 sec 5 d 8 sec 6 d 5 sin cos d 5 cos cos d 9 sec d tan 5 d tan d tan sec d 5 sin sin d 5 sin cos d

24 SECTION 8 Trigonometric Integrals 5 In Eercises 55 6, find the integral Use a computer algebra sstem to confirm our result 55 cot 56 tan d sec d 57 csc d cot t 6 csc t dt 6 6 sec tan d 6 tan t sec t dt 6 In Eercises 65 7, evaluate the definite integral 65 sin 66 d tan d 67 tan 68 d sec ttan t dt cos t 69 7 sin cos d sin t dt 7 cos d 7 In Eercises 7 78, use a computer algebra sstem to find the integral Graph the antiderivatives for two different values of the constant of integration 7 cos 7 d csc cot d cot t csc t dt sin cos d cos sec t cos t dt sin d sin cos d In Eercises 85 and 86, (a) find the indefinite integral in two different was (b) Use a graphing utilit to graph the antiderivative (without the constant of integration) obtained b each method to show that the results differ onl b a constant (c) Verif analticall that the results differ onl b a constant 85 sec tan d 86 sec tan d Area In Eercises 87 9, find the area of the region bounded b the graphs of the equations Volume In Eercises 9 and 9, find the volume of the solid generated b revolving the region bounded b the graphs of the equations about the -ais 9 Writing About Concepts (continued) 8 In our own words, describe how ou would integrate sec m tan n d for each condition (a) m is positive and even (b) n is positive and odd (c) n is positive and even, and there are no secant factors (d) sin, sin, cos, cos, tan, m is positive and odd, and there are no tangent factors sin,, sin, sin cos,,,,,, 75 sec 5 d 76 tan d 9 cos, sin,, 77 sec 5 tan d 78 sec tan d In Eercises 79 8, use a computer algebra sstem to evaluate the definite integral 79 sin sin d 8 8 sin d 8 Writing About Concepts cos d sin 6 d 8 In our own words, describe how ou would integrate sin m cos n d for each condition (a) m is positive and odd (b) n is positive and odd (c) m and n are both positive and even Volume and Centroid In Eercises 9 and 9, for the region bounded b the graphs of the equations, find (a) the volume of the solid formed b revolving the region about the -ais and (b) the centroid of the region 9 9 In Eercises 95 98, use integration b parts to verif the reduction formula sin,,, cos,,, sinn d sinn cos n cos n d cosn sin n n n sin n d n n cos n d cosm sin n d cosm sin n m n n m ncos m sin n d 98 sec n d n secn tan n n sec n d

25 5 CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals In Eercises 99, use the results of Eercises to find the integral 99 sin 5 d cos d sec sin cos d 5 d Modeling Data The table shows the normal maimum (high) and minimum (low) temperatures (in degrees Fahrenheit) for Erie, Pennslvania for each month of the ear (Source: NOAA) Month Jan Feb Mar Apr Ma Jun Ma Min The inner product of two functions f and g on a, b is given b f, g b a fg d Two distinct functions f and g are said to be orthogonal if f, g Show that the following set of functions is orthogonal on, sin, sin, sin,, cos, cos, cos, 6 Fourier Series The following sum is a finite Fourier series f N a i sin i i a sin a sin a sin a N sin N (a) Use Eercise 5 to show that the nth coefficient given b a n f sin n d (b) Let f Find a, a, and a a n is Month Jul Aug Sep Oct Nov Dec Ma Min The maimum and minimum temperatures can be modeled b t ft a a cos 6 b sin t 6 where t corresponds to Januar and a, a, and b are as follows a ft dt a 6 t ft cos 6 dt b 6 t ft sin 6 dt (a) Approimate the model Ht for the maimum temperatures (Hint: Use Simpson s Rule to approimate the integrals and use the Januar data twice) (b) Repeat part (a) for a model Lt for the minimum temperature data (c) Use a graphing utilit to compare each model with the actual data During what part of the ear is the difference between the maimum and minimum temperatures greatest? Wallis s Formulas Use the result of Eercise 96 to prove the following versions of Wallis s Formulas (a) If n is odd n, then cos n d n n (b) If n is even n, then cos n d 5 6 n n

26 SECTION 8 Trigonometric Substitution 5 Section 8 EXPLORATION Integrating a Radical Function Up to this point in the tet, ou have not evaluated the following integral d From geometr, ou should be able to find the eact value of this integral what is it? Using numerical integration, with Simpson s Rule or the Trapezoidal Rule, ou can t be sure of the accurac of the approimation Wh? Tr finding the eact value using the substitution sin and d cos d Does our answer agree with the value ou obtained using geometr? Trigonometric Substitution Use trigonometric substitution to solve an integral Use integrals to model and solve real-life applications Trigonometric Substitution Now that ou can evaluate integrals involving powers of trigonometric functions, ou can use trigonometric substitution to evaluate integrals involving the radicals a u, and The objective with trigonometric substitution is to einate the radical in the integrand You do this with the Pthagorean identities For eample, if a >, let u a sin, where Then a u a a sin a sin Note that cos sin, a cos a cos a u, cos, because sec tan, Trigonometric Substitution a > For integrals involving a u, let u a sin Then a u a cos, where For integrals involving a u, let u a tan Then a u a sec, where < < For integrals involving u a, let u a sec Then u a ±a tan, where < or < Use the positive value if u > a and the negative value if u < a u a and tan sec θ θ θ a a u a + u u a a u u u a NOTE The restrictions on ensure that the function that defines the substitution is one-to-one In fact, these are the same intervals over which the arcsine, arctangent, and arcsecant are defined

27 5 CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals EXAMPLE Trigonometric Substitution: u a sin θ 9 sin cot, Figure 86 9 Find d 9 Solution First, note that none of the basic integration rules applies To use trigonometric substitution, ou should observe that 9 is of the form a u So, ou can use the substitution a sin sin Using differentiation and the triangle shown in Figure 86, ou obtain d cos d, So, trigonometric substitution ields d 9 cos d 9 sin cos and Substitute Simplif Trigonometric identit Appl Cosecant Rule Substitute for cot Note that the triangle in Figure 86 can be used to convert the s back to s as follows cot adj opp 9 9 d sin 9csc d 9 cos, 9 cot C 9 9 C 9 C 9 9 sin Tr It Eploration A Eploration B TECHNOLOGY d 9 Then use trigonometric substitution to duplicate the results obtained with the computer algebra sstem In an earlier chapter, ou saw how the inverse hperbolic functions can be used to evaluate the integrals du u ± a, Use a computer algebra sstem to find each definite integral d d d du a u, and du ua ± u You can also evaluate these integrals using trigonometric substitution This is shown in the net eample

28 SECTION 8 Trigonometric Substitution 55 EXAMPLE Trigonometric Substitution: u a tan + Find d Solution Let u, a, and tan, as shown in Figure 87 Then, θ tan, sec Figure 87 d sec d and Trigonometric substitution produces d sec d sec sec d sec Substitute Simplif ln sec tan C ln C Appl Secant Rule Back-substitute Tr checking this result with a computer algebra sstem Is the result given in this form or in the form of an inverse hperbolic function? Tr It Eploration A Eploration B Eploration C Eploration D You can etend the use of trigonometric substitution to cover integrals involving epressions such as a u n b writing the epression as a u n a u n EXAMPLE Trigonometric Substitution: Rational Powers θ + tan, sin Figure 88 Find d Solution Begin b writing as Then, let a and u tan, as shown in Figure 88 Using d sec d and ou can appl trigonometric substitution as follows d d sec d sec d sec sec Rewrite denominator Substitute Simplif cos d sin C C Trigonometric identit Appl Cosine Rule Back-substitute Tr It Eploration A Eploration B Open Eploration

29 56 CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals For definite integrals, it is often convenient to determine the integration its for that avoid converting back to You might want to review this procedure in Section 5, Eamples 8 and 9 θ tan sec, Figure 89 EXAMPLE Converting the Limits of Integration Evaluate d Solution Because has the form u a, ou can consider u, and as shown in Figure 89 Then, and To determine the upper and lower its of integration, use the substitution sec, as follows Lower Limit a, d sec tan d sec tan Upper Limit When, sec So, ou have and When, sec and 6 Integration its for Integration its for 6 tan sec tan d d sec 6 tan d 6 sec d tan Tr It Eploration A Eploration B In Eample, tr converting back to the variable and evaluating the antiderivative at the original its of integration You should obtain d arcsec

30 SECTION 8 Trigonometric Substitution 57 When using trigonometric substitution to evaluate definite integrals, ou must be careful to check that the values of lie in the intervals discussed at the beginning of this section For instance, if in Eample ou had been asked to evaluate the definite integral d then using u and a in the interval, would impl that u < a So, when determining the upper and lower its of integration, ou would have to choose such that < In this case the integral would be evaluated as follows d 56 tan d 56 sec d tan 6 9 tan sec tan d sec 5 6 Trigonometric substitution can be used with completing the square For instance, tr evaluating the following integral d To begin, ou could complete the square and write the integral as d 56 Trigonometric substitution can be used to evaluate the three integrals listed in the following theorem These integrals will be encountered several times in the remainder of the tet When this happens, we will simpl refer to this theorem (In Eercise 85, ou are asked to verif the formulas given in the theorem) 56 THEOREM 8 Special Integration Formulas a > a u du a arcsin u ua a u C u a du uu a a lnu u a C, u a du uu a a ln u u a C u > a

31 58 CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals Applications EXAMPLE 5 Finding Arc Length (, ) The arc length of the curve from, to, Figure 8 Editable Graph f() =, Find the arc length of the graph of f from to (see Figure 8) Solution Refer to the arc length formula in Section 7 s f d d sec d sec tan ln sec tan ln 8 Formula for arc length f Let a and tan Eample 5, Section 8 Tr It EXAMPLE 6 Eploration A Comparing Two Fluid Forces A sealed barrel of oil (weighing 8 pounds per cubic foot) is floating in seawater (weighing 6 pounds per cubic foot), as shown in Figures 8 and 8 (The barrel is not completel full of oil on its side, the top foot of the barrel is empt) Compare the fluid forces against one end of the barrel from the inside and from the outside The barrel is not quite full of oil the top foot of the barrel is empt Figure 8 Rotatable Graph + = ft 8 ft Figure 8 Solution In Figure 8, locate the coordinate sstem with the origin at the center of the circle given b To find the fluid force against an end of the barrel from the inside, integrate between and 8 (using a weight of w 8) d F w hl d General equation (see Section 77) c 8 F inside 8 8 d d 96 d To find the fluid force from the outside,integrate between and (using a weight of w 6) F outside 6 d 5 d 8 d The details of integration are left for ou to complete in Eercise 8 Intuitivel, would ou sa that the force from the oil (the inside) or the force from the seawater (the outside) is greater? B evaluating these two integrals, ou can determine that F inside pounds and F outside 9 pounds Tr It Eploration A

32 SECTION 8 Trigonometric Substitution 59 Eercises for Section 8 The smbol Click on Click on indicates an eercise in which ou are instructed to use graphing technolog or a smbolic computer algebra sstem to view the complete solution of the eercise to print an enlarged cop of the graph In Eercises, use differentiation to match the antiderivative with the correct integral [Integrals are labeled (a), (b), (c), and (d)] (a) (c) 6 d 7 6 d ln 6 In Eercises 5 8, find the indefinite integral using the substitution 5 sin d d d 5 d In Eercises 9, find the indefinite integral using the substitution sec 9 d d d d (b) (d) 8 ln arcsin 8 arcsin 6 C In Eercises 6, find the indefinite integral using the substitution tan 9 d d 5 6 d In Eercises 7, use the Special Integration Formulas (Theorem 8) to find the integral 7 9 d 8 d 9 5 d d In Eercises, find the integral 9 d 9 d 6 C d 6 d C C d 6 d 5 d d 6 d t 7 8 t 9 dt 9 d 9 d d 5 6 d 9 d d 5 d 5 e e d 6 d 7 8 e e d d 9 d d arcsec d, > arcsin d In Eercises 6, complete the square and find the integral d d d In Eercises 7 5, evaluate the integral using (a) the given integration its and (b) the its obtained b trigonometric substitution 7 8 t In Eercises 5 and 5, find the particular solution of the differential equation d 9 5 d t 9 d 9 d dt d d 9,, 5 d,, d 6 5 d dt t 5

33 55 CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals In Eercises 55 58, use a computer algebra sstem to find the integral Verif the result b differentiation d 9 d d d Writing About Concepts 59 State the substitution ou would make if ou used trigonometric substitution and the integral involving the given radical, where a > Eplain our reasoning (a) a u (b) a u (c) u a 6 State the method of integration ou would use to perform each integration Eplain wh ou chose that method Do not integrate (a) d True or False? In Eercises 6 66, determine whether the statement is true or false If it is false, eplain wh or give an eample that shows it is false 6 If sin, then d d 6 If sec, then 65 If tan, then 66 If sin, then (b) d 6 Evaluate the integral using (a) u-substitution 9 d and (b) trigonometric substitution Discuss the results 6 Evaluate the integral (a) algebraicall using 9 d 9 9 and (b) using trigonometric substitution Discuss the results d sec tan d d cos d d sin cos d 68 Area Find the area of the shaded region of the circle of radius a, if the chord is h units < h < a from the center of the circle (see figure) a a a 69 Mechanical Design The surface of a machine part is the region between the graphs of and k 5 (see figure) h (, k) a (a) Find k if the circle is tangent to the graph of (b) Find the area of the surface of the machine part (c) Find the area of the surface of the machine part as a function of the radius r of the circle 7 Volume The ais of a storage tank in the form of a righ circular clinder is horizontal (see figure) The radius and length of the tank are meter and meters, respectivel m m 67 Area Find the area enclosed b the ellipse shown in the figure a b b = b a a = b a a a Rotatable Graph (a) Determine the volume of fluid in the tank as a function of its depth d (b) Use a graphing utilit to graph the function in part (a) (c) Design a dip stick for the tank with markings of,, and (d) Fluid is entering the tank at a rate of cubic meter per minute Determine the rate of change of the depth of the fluid as a function of its depth d (e) Use a graphing utilit to graph the function in part (d) When will the rate of change of the depth be minimum? Does this agree with our intuition? Eplain d

34 SECTION 8 Trigonometric Substitution 55 Volume of a Torus In Eercises 7 and 7, find the volume of the torus generated b revolving the region bounded b the graph of the circle about the -ais 7 (see figure) 8 Field Strength The field strength H of a magnet of length L on a particle r units from the center of the magnet is H ml r L Circle: ( ) + = where ±m are the poles of the magnet (see figure) Find the average field strength as the particle moves from to R units from the center b evaluating the integral R +m R ml r L dr + = L r Rotatable Graph m 7 Arc Length In Eercises 7 and 7, find the arc length of the curve over the given interval Arc Length Show that the length of one arch of the sine curve is equal to the length of one arch of the cosine curve 76 Conjecture (a) Find formulas for the distance between, and a, a along the line between these points and along the parabola (b) Use the formulas from part (a) to find the distances for a and a (c) Make a conjecture about the difference between the two distances as a increases Projectile Motion In Eercises 77 and 78, (a) use a graphing utilit to graph the path of a projectile that follows the path given b the graph of the equation, (b) determine the range of the projectile, and (c) use the integration capabilities of a graphing utilit to determine the distance the projectile travels Centroid In Eercises 79 and 8, find the centroid of the region determined b the graphs of the inequalities 79 8 h r, ln,, 7, 5, 5 h > r 9,,,, 6, 8 Surface Area Find the surface area of the solid generated b revolving the region bounded b the graphs of,,, and about the -ais Figure for 8 Figure for 8 Rotatable Graph 8 Fluid Force Find the fluid force on a circular observation window of radius foot in a vertical wall of a large water-filled tank at a fish hatcher when the center of the window is (a) fee and (b) d feet d > below the water s surface (see figure) Use trigonometric substitution to evaluate the one integral (Recal that in Section 77 in a similar problem, ou evaluated one integral b a geometric formula and the other b observing tha the integrand was odd) 8 Fluid Force Evaluate the following two integrals, which ield the fluid forces given in Eample 6 8 (a) F inside 8 8 d (b) F outside 6 d 85 Use trigonometric substitution to verif the integration formulas given in Theorem 8 86 Arc Length Show that the arc length of the graph of sin on the interval, is equal to the circumference of the ellipse (see figure) Figure for 86 Figure for Area of a Lune The crescent-shaped region bounded b two circles forms a lune (see figure) Find the area of the lune given that the radius of the smaller circle is and the radius of the larger circle is 5 5

35 55 CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals Section 85 Partial Fractions Understand the concept of a partial fraction decomposition Use partial fraction decomposition with linear factors to integrate rational functions Use partial fraction decomposition with quadratic factors to integrate rational functions Partial Fractions θ 5 sec 5 Figure This section eamines a procedure for decomposing a rational function into simpler rational functions to which ou can appl the basic integration formulas This procedure is called the method of partial fractions To see the benefit of the method of partial fractions, consider the integral 5 6 d To evaluate this integral without partial fractions, ou can complete the square and use trigonometric substitution (see Figure 8) to obtain a, 5 sec 5 6 d d 5 sec tan d tan d sec tan d csc d JOHN BERNOULLI (667 78) The method of partial fractions was introduced b John Bernoulli, a Swiss mathematician who was instrumental in the earl development of calculus John Bernoulli was a professor at the Universit of Basel and taught man outstanding students, the most famous of whom was Leonhard Euler MathBio Now, suppose ou had observed that 5 6 Then ou could evaluate the integral easil, as follows 5 6 d d Partial fraction decomposition This method is clearl preferable to trigonometric substitution However, its use depends on the abilit to factor the denominator, 5 6, and to find the partial fractions and ln ln csc cot C 5 ln C ln C ln C ln ln ln ln C 5 6 C C In this section, ou will stud techniques for finding partial fraction decompositions

36 SECTION 85 Partial Fractions 55 STUDY TIP In precalculus ou learned how to combine functions such as 5 The method of partial fractions shows ou how to reverse this process 5?? Recall from algebra that ever polnomial with real coefficients can be factored into linear and irreducible quadratic factors* For instance, the polnomial 5 can be written as 5 where is a linear factor, is a repeated linear factor, and is an irreducible quadratic factor Using this factorization, ou can write the partial fraction decomposition of the rational epression N 5 where N is a polnomial of degree less than 5, as follows N A B C D E Decomposition of N/D into Partial Fractions Divide if improper: If ND is an improper fraction (that is, if the degree of the numerator is greater than or equal to the degree of the denominator), divide the denominator into the numerator to obtain N D a polnomial N D where the degree of N is less than the degree of D Then appl Steps,, and to the proper rational epression N D Factor denominator: Completel factor the denominator into factors of the form p q m and a b c n where a b c is irreducible Linear factors: For each factor of the form p q m, the partial fraction decomposition must include the following sum of m fractions A p q A p q A m p q m Quadratic factors: For each factor of the form a b c n, the partial fraction decomposition must include the following sum of n fractions B C a b c B C a b c B n C n a b c n * For a review of factorization techniques, see Precalculus, 6th edition, b Larson and Hostetler or Precalculus: A Graphing Approach, th edition, b Larson, Hostetler, and Edwards (Boston, Massachusetts: Houghton Mifflin, and 5, respectivel)

37 55 CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals Linear Factors Algebraic techniques for determining the constants in the numerators of a partial decomposition with linear or repeated linear factors are shown in Eamples and EXAMPLE Distinct Linear Factors Write the partial fraction decomposition for Solution Because 5 6, ou should include one partial fraction for each factor and write 5 6 A where A and B are to be determined Multipling this equation b the least common denominator ) ields the basic equation A B B 5 6 Basic equation Because this equation is to be true for all, ou can substitute an convenient values for to obtain equations in A and B The most convenient values are the ones that make particular factors equal to NOTE Note that the substitutions for in Eample are chosen for their convenience in determining values for A and B; is chosen to einate the term A, and is chosen to einate the term B The goal is to make convenient substitutions whenever possible To solve for A, let and obtain A B A B A To solve for B, let and obtain A B A B B So, the decomposition is 5 6 as shown at the beginning of this section Let in basic equation Let in basic equation Tr It Eploration A Eploration B Eploration C Be sure ou see that the method of partial fractions is practical onl for integrals of rational functions whose denominators factor nicel For instance, if the denominator in Eample were changed to 5 5, its factorization as would be too cumbersome to use with partial fractions In such cases, ou should use completing the square or a computer algebra sstem to perform the integration If ou do this, ou should obtain 5 d ln ln 5 5 C

38 SECTION 85 Partial Fractions 555 EXAMPLE Repeated Linear Factors TECHNOLOGY Most computer algebra sstems, such as Derive, Maple, Mathcad, Mathematica, and the TI-89, can be used to convert a rational function to its partial fraction decomposition For instance, using Maple, ou obtain the following > convert 5 6, parfrac, 6 9 Find Solution 5 6 d Because ( ou should include one fraction for each power of and and write 5 6 A B C Multipling b the least common denominator ields the basic equation 5 6 A B C Basic equation To solve for A, let This einates the B and C terms and ields 6 A A 6 To solve for C, let This einates the A and B terms and ields 5 6 C The most convenient choices for have been used, so to find the value of B, ou can use an other value of along with the calculated values of A and C Using, A 6, and C 9 produces 5 6 A B C B So, it follows that C 9 6 B 9 B 5 6 d 6 9 d 6 ln ln 9 C ln 6 9 C Tr checking this result b differentiating Include algebra in our check, simplifing the derivative until ou have obtained the original integrand Tr It Eploration A Eploration B Eploration C NOTE It is necessar to make as man substitutions for as there are unknowns A, B, C, to be determined For instance, in Eample, three substitutions,, and were made to solve for A, B, and C

39 556 CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals Quadratic Factors When using the method of partial fractions with linear factors, a convenient choice of immediatel ields a value for one of the coefficients With quadratic factors, a sstem of linear equations usuall has to be solved, regardless of the choice of EXAMPLE Find Solution Because Distinct Linear and Quadratic Factors ou should include one partial fraction for each factor and write To solve for A, let and obtain To solve for B, let and obtain B5 ields the basic At this point, C and D are et to be determined You can find these remaining constants b choosing two other values for and solving the resulting sstem of linear equations If, then, using A and B, ou can write C D If, ou have 8 8 C D 8 C D Solving the linear sstem b subtracting the first equation from the second C D ields C Consequentl, D, and it follows that 8 d 8 A B C D Multipling b the least common denominator equation 8 A B C D 8 A C D C D 8 8 d A B d ln ln ln arctan C Tr It Eploration A Open Eploration

40 SECTION 85 Partial Fractions 557 In Eamples,, and, the solution of the basic equation began with substituting values of that made the linear factors equal to This method works well when the partial fraction decomposition involves linear factors However, if the decomposition involves onl quadratic factors, an alternative procedure is often more convenient EXAMPLE Repeated Quadratic Factors Find 8 d Solution Include one partial fraction for each power of and write 8 A B C D Multipling b the least common denominator ields the basic equation 8 A B C D Epanding the basic equation and collecting like terms produces 8 A A B B C D 8 A B A C B D Now, ou can equate the coefficients of like terms on opposite sides of the equation 8 A B D 8 A B A C B D B A C Using the known values A 8 and B, ou can write A C 8 C B D D Finall, ou can conclude that 8 d 8 ln C D d C Tr It Eploration A Technolog TECHNOLOGY Use a computer algebra sstem to evaluate the integral in Eample ou might find that the form of the antiderivative is different For instance, when ou use a computer algebra sstem to work Eample, ou obtain 8 d ln Is this result equivalent to that obtained in Eample? C

41 558 CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals When integrating rational epressions, keep in mind that for improper rational epressions such as N D 7 7 ou must first divide to obtain N 5 D The proper rational epression is then decomposed into its partial fractions b the usual methods Here are some guidelines for solving the basic equation that is obtained in a partial fraction decomposition Guidelines for Solving the Basic Equation Linear Factors Substitute the roots of the distinct linear factors into the basic equation For repeated linear factors, use the coefficients determined in guideline to rewrite the basic equation Then substitute other convenient values of and solve for the remaining coefficients Quadratic Factors Epand the basic equation Collect terms according to powers of Equate the coefficients of like powers to obtain a sstem of linear equations involving A, B, C, and so on Solve the sstem of linear equations Before concluding this section, here are a few things ou should remember First, it is not necessar to use the partial fractions technique on all rational functions For instance, the following integral is evaluated more easil b the Log Rule d ln C Second, if the integrand is not in reduced form, reducing it ma einate the need for partial fractions, as shown in the following integral d d d d ln C Finall, partial fractions can be used with some quotients involving transcendental functions For instance, the substitution u sin allows ou to write cos u sin, du cos d sin sin d du uu

42 SECTION 85 Partial Fractions 559 Eercises for Section 85 The smbol Click on Click on indicates an eercise in which ou are instructed to use graphing technolog or a smbolic computer algebra sstem to view the complete solution of the eercise to print an enlarged cop of the graph In Eercises 6, write the form of the partial fraction decomposition of the rational epression Do not solve for the constants In Eercises 7 8, use partial fractions to find the integral 7 8 d 9 d 5 d 9 6 d d 9 8 d d 9 d 8 d d d 5 9 d d 5 d d 5 d 5 5 d d 8 d d 7 d 6 9 d In Eercises 6, use substitution to find the integral sin cos cos d cos sin sin d 5 6 e e d In Eercises 7 5, use the method of partial fractions to verif the integration formula 7 a b d a ln a b C 8 a d a ln a a C 9 5 d, d, d, d, a b d a b a b d a b a ln a b C Slope Fields In Eercises 5 and 5, use a computer algebra sstem to graph the slope field for the differential equation and graph the solution through the given initial condition d 5 5 d 6 e 6,,, 6, sin cos cos d sec tan tan d a b ln a b C d d 5 e e e d In Eercises 9, evaluate the definite integral Use a graphing utilit to verif our result 9 5 d d In Eercises, use a computer algebra sstem to determine the antiderivative that passes through the given point Use the sstem to graph the resulting antiderivative, 6 9 d, 5, 6 d, 5 d d 6 d, d,,, Writing About Concepts 5 What is the first step when integrating Eplain 5 d? 5 Describe the decomposition of the proper rational function ND (a) if D p q m, and (b) if D a b c n, where a b c is irreducible Eplain wh ou chose that method 55 State the method ou would use to evaluate each integral Eplain wh ou chose that method Do not integrate (a) (b) 7 (c) 8 d 8 d 5 d

43 56 CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals Writing About Concepts (continued) 56 Determine which value best approimates the area of the region between the -ais and the graph of f over the interval, Make our selection on the basis of a sketch of the region and not b performing an calculations Eplain our reasoning 57 Area Find the area of the region bounded b the graphs of 5 6,,, and 58 Area Find the area of the region bounded b the graphs of 76 and 59 Modeling Data The predicted cost C (in hundreds of thousands of dollars) for a compan to remove p% of a chemical from its waste water is shown in the table A model for the data is given b Use the model to find the average cost for removing between 75% and 8% of the chemical 6 Logistic Growth In Chapter 6, the eponential growth equation was derived from the assumption that the rate of growth was proportional to the eisting quantit In practice, there often eists some upper it L past which growth cannot occur In such cases, ou assume the rate of growth to be proportional not onl to the eisting quantit, but also to the difference between the eisting quantit and the upper it L That is, ddt kl In integral form, ou can write this relationship as (a) 6 (b) 6 (c) (d) 5 (e) 8 p C C p p p, d L k dt (a) A slope field for the differential equation ddt is shown Draw a possible solution to the differential equation if 5, and another if To print an enlarged cop of the graph, select the MathGraph button 5 p < (b) Where is greater than, what is the sign of the slope of the solution? (c) For >, find t t (d) Evaluate the two given integrals and solve for as a function of t, where is the initial quantit (e) Use the result of part (d) to find and graph the solutions in part (a) Use a graphing utilit to graph the solutions and compare the results with the solutions in part (a) (f) The graph of the function is a logistic curve Show tha the rate of growth is maimum at the point of inflection and that this occurs when L 6 Volume and Centroid Consider the region bounded b the graphs of,,, and Find the volume of the solid generated b revolving the region abou the -ais Find the centroid of the region 6 Volume Consider the region bounded b the graph of on the interval, Find the volume of the solid generated b revolving this region about the -ais 6 Epidemic Model A single infected individual enters a communit of n susceptible individuals Let be the number of newl infected individuals at time t The common epidemic model assumes that the disease spreads at a rate proportional to the product of the total number infected and the number not e infected So, ddt k n and ou obtain Solve for as a function of t 6 Chemical Reactions In a chemical reaction, one unit of compound Y and one unit of compound Z are converted into a single unit of compound X is the amount of compound X formed, and the rate of formation of X is proportional to the product of the amounts of unconverted compounds Y and Z So ddt k z, where and z are the initia amounts of compounds Y and Z From this equation ou obtain (a) Perform the two integrations and solve for in terms of t (b) Use the result of part (a) to find as t if () < z () > z, and () z 65 Evaluate n d k dt z d k dt d in two different was, one of which is partial fractions 5 t 66 Prove Putnam Eam Challenge 7 d This problem was composed b the Committee on the Putnam Prize Competition The Mathematical Association of America All rights reserved

44 SECTION 86 Integration b Tables and Other Integration Techniques 56 Section 86 Integration b Tables and Other Integration Techniques Evaluate an indefinite integral using a table of integrals Evaluate an indefinite integral using reduction formulas Evaluate an indefinite integral involving rational functions of sine and cosine Integration b Tables So far in this chapter ou have studied several integration techniques that can be used with the basic integration rules But merel knowing how to use the various techniques is not enough You also need to know when to use them Integration is first and foremost a problem of recognition That is, ou must recognize which rule or technique to appl to obtain an antiderivative Frequentl, a slight alteration of an integrand will require a different integration technique (or produce a function whose antiderivative is not an elementar function), as shown below TECHNOLOGY A computer algebra sstem consists, in part, of a database of integration formulas The primar difference between using a computer algebra sstem and using tables of integrals is that with a computer algebra sstem the computer searches through the database to find a fit With integration tables, ou must do the searching ln d ln C ln ln d C Integration b parts Power Rule Log Rule Not an elementar function Man people find tables of integrals to be a valuable supplement to the integration techniques discussed in this chapter Tables of common integrals can be found in Appendi B Integration b tables is not a cure-all for all of the difficulties that can accompan integration using tables of integrals requires considerable thought and insight and often involves substitution Each integration formula in Appendi B can be developed using one or more of the techniques in this chapter You should tr to verif several of the formulas For instance, Formula ln d ln ln C d? ln u a bu du b Formula can be verified using the method of partial fractions, and Formula 9 a bu du du a bu a Formula 9 u ua bu can be verified using integration b parts Note that the integrals in Appendi B are classified according to forms involving the following u ± a u n a bu a bu cu a bu a ± u a u Trigonometric functions Inverse trigonometric functions Logarithmic functions a a bu ln a bu C Eponential functions

45 56 CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals EXPLORATION Use the tables of integrals in Appendi B and the substitution u to evaluate the integral in Eample If ou do this, ou should obtain d du u Does this produce the same result as that obtained in Eample? EXAMPLE Find Integration b Tables Solution Because the epression inside the radical is linear, ou should consider forms involving a bu du Formula 7 a < ua bu a arctan a bu C a Let a, b, and u Then du d, and ou can write d d arctan C Tr It Eploration A EXAMPLE Integration b Tables Find 9 d Solution Because the radical has the form u a, ou should consider Formula 6 u a du uu a a lnu u a C Let u and a Then du d, and ou have 9 d d 9 9 ln 9 C Tr It Eploration A Open Eploration EXAMPLE Find Integration b Tables Solution Of the forms involving e u, consider the following formula e d du e u u ln eu C Let u Then du d, and ou have d e d e ln e C Formula 8 ln e C Tr It Eploration A TECHNOLOGY Eample shows the importance of having several solution techniques at our disposal This integral is not difficult to solve with a table, but when it was entered into a well-known computer algebra sstem, the utilit was unable to find the antiderivative

46 SECTION 86 Integration b Tables and Other Integration Techniques 56 Reduction Formulas Several of the integrals in the integration tables have the form f d g h d Such integration formulas are called reduction formulas because the reduce a given integral to the sum of a function and a simpler integral EXAMPLE Using a Reduction Formula Find sin d Solution Consider the following three formulas u sin u du sin u u cos u C Formula 5 TECHNOLOGY Sometimes when ou use computer algebra sstems ou obtain results that look ver different, but are actuall equivalent Here is how several different sstems evaluated the integral in Eample 5 Maple 5 arctanh 5 Derive 5 ln 5 Mathematica Sqrt 5 Sqrt 5 Sqrt ArcTanh Sqrt Mathcad 5 ln Notice that computer algebra sstems do not include a constant of integration u n sin u du u n cos u n u n cos u du u n cos u du u n sin u n u n sin u du Using Formula 5, Formula 55, and then Formula 5 produces EXAMPLE 5 Using a Reduction Formula 5 Find d Solution sin d cos cos d Tr It Consider the following two formulas du ua bu a ln a bu a a bu C a bu du du a bu a u ua bu Using Formula 9, with a, b 5, and u, produces 5 cos sin sin d d 5 5 d 5 d 5 Formula 5 Formula 55 cos sin 6 cos 6 sin C Eploration A Formula 7 a > Formula 9 Using Formula 7, with a, b 5, and u, produces 5 d 5 ln 5 5 C 5 ln 5 C 5 Tr It Eploration A Eploration B

47 56 CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals Rational Functions of Sine and Cosine EXAMPLE 6 Find sin cos d Integration b Tables Solution Substituting sin cos for sin produces sin cos d sin cos cos d A check of the forms involving sin u or cos u in Appendi B shows that none of those listed applies So, ou can consider forms involving a bu For eample, u du a bu b bu a ln a bu C Formula Let a, b, and u cos Then du sin d, and ou have sin cos cos d cos sin d cos cos ln cos ln cos C cos C Tr It Eploration A Eample 6 involves a rational epression of sin and cos If ou are unable to find an integral of this form in the integration tables, tr using the following special substitution to convert the trigonometric epression to a standard rational epression Substitution for Rational Functions of Sine and Cosine For integrals involving rational functions of sine and cosine, the substitution ields u sin cos tan u cos u, sin u u, and d du u Proof From the substitution for u, it follows that sin u cos cos cos cos cos Solving for cos produces cos u u To find sin, write u sin cos as sin u cos u u u u Finall, to find d, consider u tan Then ou have arctan u and d du u

48 SECTION 86 Integration b Tables and Other Integration Techniques 565 Eercises for Section 86 The smbol Click on Click on indicates an eercise in which ou are instructed to use graphing technolog or a smbolic computer algebra sstem to view the complete solution of the eercise to print an enlarged cop of the graph In Eercises and, use a table of integrals with forms involving a bu to find the integral d In Eercises and, use a table of integrals with forms involving u ± a to find the integral 9 e e d d In Eercises 5 and 6, use a table of integrals with forms involving a u to find the integral 5 6 d In Eercises 7, use a table of integrals with forms involving the trigonometric functions to find the integral cos 7 8 sin d d 9 cos d In Eercises and, use a table of integrals with forms involving e u to find the integral d e sin d e In Eercises and, use a table of integrals with forms involving ln u to find the integral ln d ln d In Eercises 5 8, find the indefinite integral (a) using integration tables and (b) using the given method 5 e d Integration b parts 6 ln d Integration b parts Integral 7 Partial fractions d 8 Partial fractions 75 d In Eercises 9, use integration tables to find the integral 9 arcsec d arcsec d Method 5 d 9 d tan 5 d d d d sin d e 5 e 6 arccos e d tan e d 7 8 sec d t ln t cos 9 dt 9 d sin sin arctan d ln e d 9 d e d ln d 9 d e tan d e d In Eercises 5, use integration tables to evaluate the integral e d d 5 6 ln d sin d cos 7 8 sin d 5 d 9 t cos t dt 5 In Eercises 5 56, verif the integration formula d d d d u b bu a du a bu u n du a bu u ± a du ±u a u ± a C u n cos u du u n sin u n u n sin u du a bu a ln a bu C n b un a bu na un 55 arctan u du u arctan u ln u C cos sin d d a bu du

49 566 CHAPTER 8 Integration Techniques, L Hôpital s Rule, and Improper Integrals 56 In Eercises 57 6, use a computer algebra sstem to determine the antiderivative that passes through the given point Use the sstem to graph the resulting antiderivative 57 58, 5 d,, d, ln u n du uln u n n ln u n du, 6 d, d,, In Eercises 6 7, find or evaluate the integral 6 6 sin sin cos d, sin tan sin cos cos 69 7 d d d, sin, cos sin d, d sin cos cos cos cos d d sec tan d d True or False? In Eercises 8 and 8, determine whether the statement is true or false If it is false, eplain wh or give an eample that shows it is false 8 To use a table of integrals, the integral ou are evaluating mus appear in the table 8 When using a table of integrals, ou ma have to make substitutions to rewrite our integral in the form in which it appears in the table 8 Work A hdraulic clinder on an industrial machine pushes a steel block a distance of feet 5, where the variable force required is F e pounds Find the work done in pushing the block the full 5 feet through the machine 8 Work Repeat Eercise 8, using F 5 pounds 6 85 Building Design The cross section of a precast concrete beam for a building is bounded b the graphs of the equations,,, and where and are measured in feet The length of the beam is feet (see figure) (a) Find the volume V and the weight W of the beam Assume the concrete weighs 8 pounds per cubic foot (b) Then find the centroid of a cross section of the beam Area In Eercises 7 and 7, find the area of the region bounded b the graphs of the equations 7,, 8 7,, e ft Writing About Concepts In Eercises 7 78, state (if possible) the method or integration formula ou would use to find the antiderivative Eplain wh ou chose that method or formula Do not integrate 7 7 e e e e d d 75 e d 76 e d 77 e d 78 e e d 79 (a) Evaluate n ln d for n,, and Describe an patterns ou notice (b) Write a general rule for evaluating the integral in part (a), for an integer n 8 Describe what is meant b a reduction formula Give an eample 86 Population A population is growing according to the logistic model N where t is the time in das Find the average population over the interval, In Eercises 87 and 88, use a graphing utilit to (a) solve the integral equation for the constant k and (b) graph the region whose area is given b the integral k k 87 d 88 6 e d 5 89 Evaluate Rotatable Graph 5 e 89t Putnam Eam Challenge d tan This problem was composed b the Committee on the Putnam Prize Competition The Mathematical Association of America All rights reserved