Fitting Integrals to Basic Rules Basic Integration Rules Lesson 8.1 Consider these similar integrals Which one uses The log rule The arctangent rule The rewrite with long division principle Try It Out Decide which principle to apply Consider The Log Rule in Disguise The quotient suggests possible Log Rule, but the is not present We can manipulate this to make the Log Rule apply Add and subtract e x in the numerator The Power Rule in Disguise The Power Rule in Disguise Here's another integral that doesn't seem to fit the basic options Then becomes and applies What are the options for u? Best choice is Note review of basic integration rules pg 520 Note procedures for fitting integrands to basic rules, pg 521 1
Disguises with Trig Identities What rules might this fit? Note that tan 2 u is However sec 2 uison the list This suggests one of the identities and we have Lesson 8.1 Page 524 Exercises 1 49 EOO Review Product Rule Integration by Parts Recall definition of derivative of the product of two functions Lesson 8.2 Now we will manipulate this to get Manipulating the Product Rule Integration by Parts Now take the integral of both sides Which term above can be simplified? It is customary to write this using substitution u = f(x) du = v = g(x) = g'(x) dx This gives us 2
Strategy an integral we split the integrand into two parts First part labeled u The other labeled dv Guidelines for making the split Note: a certain amount of trial and error will happen in making this split The dv always includes the The must be integratable v duis than u dv Making the Split A table to keep things organized is helpful u dv du v Decide what will be the and the This determines the duand the v Now rewrite Strategy Hint Trick is to select the correct function for u A rule of thumb is the LIATEhierarchy rule The ushould be first available from L Inverse trigonometric A Trigonometric E Try This Choose a u u du and dv dv v Determine the v and the du Substitute the values, finish integration Double Trouble Going in Circles Sometimes the second integral must alsobe done by parts u x 2 du 2x dx dv sin x v -cos x When we end up with the same as we started with Try Should end up with the u dv du v Add the integral to both sides 3
Lesson 8.2A Page 533 Exercises 1 37 odd Tabular Method Works well for integrals involving Powers of x and Sin, Cos, or Example Alt Signs uand derivatives v and antiderivatives + sin 4x - 2x -1/4 cos 4x + 2-1/16 sin 4x - 0 1/64 cos 4x Application Consider the region bounded by y = cos x, y = 0, x = 0, and x = ½ π What is the volume generated by rotating the region around the y-axis? What is the radius? What is the disk thickness? What are the limits? Lesson 8.2B Page 533 Exercises 47 73 odd 103 a, b, c, 104 a, b Partial Fraction Decomposition Consider adding two algebraic fractions Partial Fractions Lesson 8.5 Partial fraction decomposition the process 4
Partial Fraction Decomposition The Process Motivation for this process The separate terms are Where polynomial P(x) has P(r) 0 Then f(x) can be decomposed with this cascading form Strategy Given N(x)/D(x) 1.If degree of N(x) degree of D(x) divide the denominator into the numerator to obtain Degree of N 1 (x) will be that of D(x) Now proceed with following steps for N 1 (x)/d(x) Strategy 2. Factor the denominator into factors of the form where is irreducible 3. For each factor the partial fraction must include the following sum of m fractions Strategy 4. Quadratic factors: For each factor of the form, the partial fraction decomposition must include the following sum of n fractions. Suppose rational function has distinct linear factors Then we know A Variation 5
A Variation Now multiply through by the denominator to clear them from the equation Let x = 1 and x = -1 (Why these values?) Solve for A and B Single irreducible quadratic factor What If But P(x) degree < 2m Then cascading form is Gotta Try It Gotta Try It Then Now equate corresponding coefficients on each side Solve for A, B, C, and D? Even More Exciting Combine the Methods When but Consider where P(x) and D(x) are polynomials with D(x) 0 Example P(x), D(x) have no common factors D(x) 0 Express as functions of 6
Try It This Time Partial Fractions for Integration Use these principles for the following integrals Now manipulate the expression to determine A, B, and C Why Are We Doing This? Remember, the whole idea is to make the rational function easier to integrate Lesson 8.5 Page 561 Exercises 1 45 EOO Tables of Integrals Integration by Tables Lesson 8.6 Text has covered only limited variety of integrals Applications in real life encounter many other types to memorize all types Tables of integrals have been established Text includes list in Appendix B, pg A-18 7
General Table Classifications Elementary forms Forms involving Forms involving Forms involving Trigonometric forms Inverse trigonometric forms Exponential, logarithmic forms Hyperbolic forms Finding the Right Form For each integral Determine the classification Use the given pattern to complete the integral Reduction Formulas Some integral patterns in the tables have the form This reduces a given integral to the sum of a and a integral This gives you Reduction Formulas Now use formula 17 Use formula 19 first of all and finish the integration Lesson 8.6 Page 567 Exercises 1 49 EOO Improper Integrals Lesson 8.8 8
Improper Integrals Note the graph of y = x -2 We seek the area under the curve to the right of x = 1 To Infinity and Beyond To solve we write as a limit (if the limit exists) Thus the integral is Known as an improper integral Evaluating Improper Integrals To Limit Or Not to Limit The limit may not exist Take the integral Consider Apply the limit Rewrite as a limit and evaluate For To Converge Or Not Try this one Improper Integral to - A limit exists (the proper integral converges) for The integral for p 1 Rewrite as a limit, integrate 9
When f(x) Unbounded at x = c When vertical asymptote exists at x = c As before, set a limit and evaluate In this case the limit is Consider Using L'Hopital's Rule Start with integration by parts dv and u = Now apply the definition of an improper integral We have Using L'Hopital's Rule Lesson 8.8 Page 587 Exercises 1 53 EOO Now use for the first term 10