Integration by Substitution. Pattern Recognition


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1 SECTION Integrtion b Substitution 9 Section Integrtion b Substitution Use pttern recognition to find n indefinite integrl Use chnge of vribles to find n indefinite integrl Use the Generl Power Rule for Integrtion to find n indefinite integrl Use chnge of vribles to evlute definite integrl Evlute definite integrl involving n even or odd function Pttern Recognition In this section ou will stud techniques for integrting composite functions The discussion is split into two prts pttern recognition nd chnge of vribles Both techniques involve usubstitution With pttern recognition ou perform the substitution mentll, nd with chnge of vribles ou write the substitution steps The role of substitution in integrtion is comprble to the role of the Chin Rule in differentition Recll tht for differentible functions given b Fu nd u g, the Chin Rule sttes tht d Fg Fgg d From the definition of n ntiderivtive, it follows tht Fgg d Fg C Fu C These results re summrized in the following theorem NOTE The sttement of Theorem doesn t tell how to distinguish between fg nd g in the integrnd As ou become more eperienced t integrtion, our skill in doing this will increse Of course, prt of the ke is fmilirit with derivtives THEOREM Antidifferentition of Composite Function Let g be function whose rnge is n intervl I, nd let f be function tht is continuous on I If g is differentible on its domin nd F is n ntiderivtive of f on I, then fgg d Fg C If u g, then du g d nd fu du Fu C EXPLORATION STUDY TIP There re severl techniques for ppling substitution, ech differing slightl from the others However, ou should remember tht the gol is the sme with ever technique ou re tring to find n ntiderivtive of the integrnd Recognizing Ptterns The integrnd in ech of the following integrls fits the pttern fgg Identif the pttern nd use the result to evlute the integrl d b c sec d tn d The net three integrls re similr to the first three Show how ou cn multipl nd divide b constnt to evlute these integrls d d e d f sec (tn d
2 96 CHAPTER Integrtion Emples nd show how to ppl Theorem directl, b recognizing the presence of fg nd g Note tht the composite function in the integrnd hs n outside function f nd n inside function g Moreover, the derivtive g is present s fctor of the integrnd Outside function fgg d Fg C Inside function Derivtive of inside function EXAMPLE Recognizing the f gg Pttern Find d TECHNOLOGY Tr using computer lgebr sstem, such s Mple, Derive, Mthemtic, Mthcd, or the TI89, to solve the integrls given in Emples nd Do ou obtin the sme ntiderivtives tht re listed in the emples? Solution Letting g, ou obtin nd g fg f From this, ou cn recognize tht the integrnd follows the fgg pttern Using the Power Rule for Integrtion nd Theorem, ou cn write fg g d C Tr using the Chin Rule to check tht the derivtive of integrnd of the originl integrl ) C is the Tr It Eplortion A Technolog Video EXAMPLE Recognizing the f gg Pttern Find cos d Solution Letting g, ou obtin nd g fg f cos From this, ou cn recognize tht the integrnd follows the fgg pttern Using the Cosine Rule for Integrtion nd Theorem, ou cn write fg g cos d sin C You cn check this b differentiting sin C to obtin the originl integrnd Tr It Eplortion A
3 SECTION Integrtion b Substitution 97 The integrnds in Emples nd fit the fgg pttern ectl ou onl hd to recognize the pttern You cn etend this technique considerbl with the Constnt Multiple Rule kf d kf d Mn integrnds contin the essentil prt (the vrible prt) of g but re missing constnt multiple In such cses, ou cn multipl nd divide b the necessr constnt multiple, s shown in Emple EXAMPLE Multipling nd Dividing b Constnt Find d Solution This is similr to the integrl given in Emple, ecept tht the integrnd is missing fctor of Recognizing tht is the derivtive of, ou cn let g nd suppl the s follows d d fg g d C 6 C Multipl nd divide b Constnt Multiple Rule Integrte Simplif Tr It Eplortion A Eplortion B In prctice, most people would not write s mn steps s re shown in Emple For instnce, ou could evlute the integrl b simpl writing d d C 6 C NOTE Be sure ou see tht the Constnt Multiple Rule pplies onl to constnts You cnnot multipl nd divide b vrible nd then move the vrible outside the integrl sign For instnce, d d After ll, if it were legitimte to move vrible quntities outside the integrl sign, ou could move the entire integrnd out nd simplif the whole process But the result would be incorrect
4 98 CHAPTER Integrtion Chnge of Vribles With forml chnge of vribles, ou completel rewrite the integrl in terms of u nd du (or n other convenient vrible) Although this procedure cn involve more written steps thn the pttern recognition illustrted in Emples to, it is useful for complicted integrnds The chnge of vrible technique uses the Leibniz nottion for the differentil Tht is, if u g, then du g d, nd the integrl in Theorem tkes the form fgg d fu du Fu C EXAMPLE Chnge of Vribles Find d Solution First, let u be the inner function, u Then clculte the differentil du to be du d Now, using u nd d du, substitute to obtin d u du Integrl in terms of u STUDY TIP Becuse integrtion is usull more difficult thn differentition, ou should lws check our nswer to n integrtion problem b differentiting For instnce, in Emple ou should differentite C to verif tht ou obtin the originl integrnd Tr It u du u C u C C Eplortion A Constnt Multiple Rule Antiderivtive in terms of u Simplif Antiderivtive in terms of EXAMPLE Chnge of Vribles Find d Solution As in the previous emple, let u nd obtin d du Becuse the integrnd contins fctor of, ou must lso solve for in terms of u, s shown u Now, using substitution, ou obtin u d u u du u u du Solve for in terms of u u u C 6 C Tr It Eplortion A Eplortion B Open Eplortion
5 SECTION Integrtion b Substitution 99 To complete the chnge of vribles in Emple, ou solved for in terms of u Sometimes this is ver difficult Fortuntel it is not lws necessr, s shown in the net emple EXAMPLE 6 Chnge of Vribles Find sin cos d STUDY TIP When mking chnge of vribles, be sure tht our nswer is written using the sme vribles s in the originl integrnd For instnce, in Emple 6, ou should not leve our nswer s 9 u C but rther, replce u b sin Solution Becuse sin sin, ou cn let u sin Then du cos d Now, becuse cos d is prt of the originl integrl, ou cn write du cos d Substituting u nd du in the originl integrl ields sin cos d u du u du u C 9 sin C You cn check this b differentiting d d 9 sin 9 sin cos sin cos Becuse differentition produces the originl integrnd, ou know tht ou hve obtined the correct ntiderivtive Tr It Eplortion A The steps used for integrtion b substitution re summrized in the following guidelines Guidelines for Mking Chnge of Vribles Choose substitution u g Usull, it is best to choose the inner prt of composite function, such s quntit rised to power Compute du g d Rewrite the integrl in terms of the vrible u Find the resulting integrl in terms of u Replce u b g to obtin n ntiderivtive in terms of 6 Check our nswer b differentiting
6 CHAPTER Integrtion The Generl Power Rule for Integrtion One of the most common usubstitutions involves quntities in the integrnd tht re rised to power Becuse of the importnce of this tpe of substitution, it is given specil nme the Generl Power Rule for Integrtion A proof of this rule follows directl from the (simple) Power Rule for Integrtion, together with Theorem THEOREM The Generl Power Rule for Integrtion If g is differentible function of, then g g n n g d C, n Equivlentl, if u g, then n u n du un C, n n EXAMPLE 7 Substitution nd the Generl Power Rule u du u b c d d C d d C u d d C C u u du u du u du u Suppose ou were sked to find one of the following integrls Which one would ou choose? Eplin our resoning EXPLORATION d or d b tn sec d or tn d d e d d C C cos cos sin d cos sin d C Some integrls whose integrnds involve quntities rised to powers cnnot be found b the Generl Power Rule Consider the two integrls d nd u du d The substitution u works in the first integrl but not in the second In the second, the substitution fils becuse the integrnd lcks the fctor needed for du Fortuntel, for this prticulr integrl, ou cn epnd the integrnd s nd use the (simple) Power Rule to integrte ech term u Tr It Eplortion A Eplortion B
7 SECTION Integrtion b Substitution Chnge of Vribles for Definite Integrls When using usubstitution with definite integrl, it is often convenient to determine the limits of integrtion for the vrible u rther thn to convert the ntiderivtive bck to the vrible nd evlute t the originl limits This chnge of vribles is stted eplicitl in the net theorem The proof follows from Theorem combined with the Fundmentl Theorem of Clculus THEOREM Chnge of Vribles for Definite Integrls If the function u g hs continuous derivtive on the closed intervl, b nd f is continuous on the rnge of g, then b gb fgg d fu du g EXAMPLE 8 Chnge of Vribles Evlute d Solution To evlute this integrl, let u Then, ou obtin u du d Before substituting, determine the new upper nd lower limits of integrtion Lower Limit When, u Now, ou cn substitute to obtin d u du u 8 Integrtion limits for Integrtion limits for u Tr rewriting the ntiderivtive u in terms of the vrible nd evlute the definite integrl t the originl limits of integrtion, s shown u d Upper Limit When, u 8 Notice tht ou obtin the sme result Tr It Eplortion A Video
8 CHAPTER Integrtion EXAMPLE 9 Evlute Chnge of Vribles Solution To evlute this integrl, let u Then, ou obtin u A u u u du d d Differentite ech side Before substituting, determine the new upper nd lower limits of integrtion (, ) = (, ) The region before substitution hs n re 6 of Figure 7 f(u) (, ) f(u) = u + (, ) The region fter substitution hs n re 6 of Figure 8 u Lower Limit When, u Now, substitute to obtin d u u u du u du Geometricll, ou cn interpret the eqution d u du u u 9 6 Upper Limit When, u Tr It Eplortion A Eplortion B Video to men tht the two different regions shown in Figures 7 nd 8 hve the sme re When evluting definite integrls b substitution, it is possible for the upper limit of integrtion of the uvrible form to be smller thn the lower limit If this hppens, don t rerrnge the limits Simpl evlute s usul For emple, fter substituting u in the integrl d ou obtin u when, nd u when So, the correct uvrible form of this integrl is u u du
9 SECTION Integrtion b Substitution Integrtion of Even nd Odd Functions Even with chnge of vribles, integrtion cn be difficult Occsionll, ou cn simplif the evlution of definite integrl (over n intervl tht is smmetric bout the is or bout the origin) b recognizing the integrnd to be n even or odd function (see Figure 9) Even function THEOREM Integrtion of Even nd Odd Functions Let f be integrble on the closed intervl, If f is n even function, then f d f d If f is n odd function, then f d Proof Becuse f is even, ou know tht f f Using Theorem with the substitution u produces f d fudu f u du f u du f d Odd function Figure 9 Finll, using Theorem 6, ou obtin f d f d f d f d f d f d This proves the first propert The proof of the second propert is left to ou (see Eercise ) Becuse f is n odd function, f d Figure f() = sin cos + sin cos Editble Grph EXAMPLE Integrtion of n Odd Function Evlute sin cos sin cos d Solution Letting f sin cos sin cos produces f sin cos sin cos sin cos sin cos f So, f is n odd function, nd becuse f is smmetric bout the origin over,, ou cn ppl Theorem to conclude tht sin cos sin cos d Tr It Eplortion A NOTE From Figure ou cn see tht the two regions on either side of the is hve the sme re However, becuse one lies below the is nd one lies bove it, integrtion produces cncelltion effect (More will be sid bout this in Section 7)
10 CHAPTER Integrtion Eercises for Section The smbol Click on Click on indictes n eercise in which ou re instructed to use grphing technolog or smbolic computer lgebr sstem to view the complete solution of the eercise to print n enlrged cop of the grph In Eercises 6, complete the tble b identifing u nd du for the integrl 6 fgg d d d d sec tn d tn sec d cos sin d u g du g d In Eercises 7, find the indefinite integrl nd check the result b differentition In Eercises 8, solve the differentil eqution d d 6 d d d d d d 8 Slope Fields In Eercises 9, differentil eqution, point, nd slope field re given A slope field consists of line segments with slopes given b the differentil eqution These line segments give visul perspective of the directions of the solutions of the differentil eqution () Sketch two pproimte solutions of the differentil eqution on the slope field, one of which psses through the given point (To print n enlrged cop of the grph, select the MthGrph button) (b) Use integrtion to find the prticulr solution of the differentil eqution nd use grphing utilit to grph the solution Compre the result with the sketches in prt () 7 d d d d tt dt 6 7 d 8 9 d d d t t dt 6 7 d d t t t dt 9 d 9 d d d d t t dt uu du d 6 d d d t t dt t t t d dt 8 d d 9 d d d cos sec tn d d,,, d d,
11 SECTION Integrtion b Substitution In Eercises 6, find the indefinite integrl sin d sin d sin d 6 cos 6 d 7 cos 8 sin d d 9 sin cos d sec tn d 7 76 d d 79 d cos d cos d d d d tn sec d csc cot d cot d 6 tn sec d sin cos d csc d Differentil Equtions In Eercises 8 86, the grph of function f is shown Use the differentil eqution nd the given point to find n eqution of the function d d 8 8 d 8 d 8 In Eercises 7 6, find n eqution for the function f tht hs the given derivtive nd whose grph psses through the given point Derivtive f cos f sec tn f sin In Eercises 6 7, find the indefinite integrl b the method shown in Emple d d 69 7 t t dt ) d In Eercises 7 8, evlute the definite integrl Use grphing utilit to verif our result 7 d 7 7 d 7 Point,, 6 f sec, 6 f, 6 f 8, 7 6 d 6 d 6 d 66 d, 8 d d d 8 86 d In Eercises 87 9, find the re of the region Use grphing utilit to verif our result 7 87 d f (, ) f (, ) d d d 8 6 f (, ) 6 f 6 (, )
12 6 CHAPTER Integrtion 89 sin sin sec d sin cos csc cot d Use the smmetr of the grphs of the sine nd cosine functions s n id in evluting ech definite integrl () (c) In Eercises 7 nd 8, write the integrl s the sum of the integrl of n odd function nd the integrl of n even function Use this simplifiction to evlute the integrl sin d cos d (b) (d) 7 6 d 8 Writing About Concepts 9 Describe wh d u du cos d sin cos d sin cos d In Eercises 9 98, use grphing utilit to evlute the integrl Grph the region whose re is given b the definite integrl 9 9 d 9 d cos 98 6 d Writing In Eercises 99 nd, find the indefinite integrl in two ws Eplin n difference in the forms of the nswers 99 d sin cos d In Eercises, evlute the integrl using the properties of even nd odd functions s n id d sin cos d Use d 8 to evlute ech definite integrl without using the Fundmentl Theorem of Clculus () (c) 7 d d (b) (d) d d d d sin d sin cos d d where u Without integrting, eplin wh d Csh Flow The rte of disbursement dqdt of million dollr federl grnt is proportionl to the squre of t Time t is mesured in ds t, nd Q is the mount tht remins to be disbursed Find the mount tht remins to be disbursed fter ds Assume tht ll the mone will be disbursed in ds Deprecition The rte of deprecition dvdt of mchine is inversel proportionl to the squre of t, where V is the vlue of the mchine t ers fter it ws purchsed The initil vlue of the mchine ws $,, nd its vlue decresed $, in the first er Estimte its vlue fter ers Rinfll The norml monthl rinfll t the SettleTcom irport cn be pproimted b the model R 99 sint 77 where R is mesured in inches nd t is the time in months, with t corresponding to Jnur (Source: US Ntionl Ocenic nd Atmospheric Administrtion) () Determine the etrem of the function over oneer period (b) Use integrtion to pproimte the norml nnul rinfll (Hint: Integrte over the intervl, ) (c) Approimte the verge monthl rinfll during the months of October, November, nd December
13 SECTION Integrtion b Substitution 7 Sles The sles S (in thousnds of units) of sesonl product re given b the model t S 7 7 sin 6 where t is the time in months, with t corresponding to Jnur Find the verge sles for ech time period () The first qurter t (b) The second qurter t 6 (c) The entire er t Wter Suppl A model for the flow rte of wter t pumping sttion on given d is Rt 7 sin t where t R is the flow rte in thousnds of gllons per hour, nd t is the time in hours () Use grphing utilit to grph the rte function nd pproimte the mimum flow rte t the pumping sttion (b) Approimte the totl volume of wter pumped in d 6 Electricit The oscillting current in n electricl circuit is where I is mesured in mperes nd t is mesured in seconds Find the verge current for ech time intervl () t 6 (b) t (c) t Probbilit f k n m, In Eercises 7 nd 8, the function where n >, m >, nd k is constnt, cn be used to represent vrious probbilit distributions If k is chosen such tht f d the probbilit tht will fll between nd b b is b P, b f d 7 The probbilit tht person will remember between % nd b% of mteril lerned in n eperiment is b P, b d cos I sin6t cost t 89 where represents the percent remembered (See figure) () For rndoml chosen individul, wht is the probbilit tht he or she will recll between % nd 7% of the mteril? (b) Wht is the medin percent recll? Tht is, for wht vlue of b is it true tht the probbilit of reclling to b is? Figure for 7 8 The probbilit tht ore smples tken from region contin between % nd b% iron is b P, b d where represents the percent of iron (See figure) Wht is the probbilit tht smple will contin between () % nd % iron? (b) % nd % iron? 9 Temperture The temperture in degrees Fhrenheit in house is T 7 sin t 8 where t is time in hours, with t representing midnight The hourl cost of cooling house is $ per degree () Find the cost C of cooling the house if its thermostt is set t 7F b evluting the integrl t 8 C 7 sin 7 (See figure) 8 dt Temperture (in F) P, b b T P, b b Thermostt setting: Time (in hours) t
14 8 CHAPTER Integrtion (b) Find the svings from resetting the thermostt to 78F b evluting the integrl 8 t 8 C 7 sin 78 dt Temperture (in F) (See figure) T Mnufcturing A mnufcturer of fertilizer finds tht ntionl sles of fertilizer follow the sesonl pttern F, sin t 6 6 where F is mesured in pounds nd t represents the time in ds, with t corresponding to Jnur The mnufcturer wnts to set up schedule to produce uniform mount of fertilizer ech d Wht should this mount be? Grphicl Anlsis Consider the functions f nd g, where t f 6 sin cos nd gt f d () Use grphing utilit to grph f nd g in the sme viewing window (b) Eplin wh g is nonnegtive (c) Identif the points on the grph of g tht correspond to the etrem of f (d) Does ech of the zeros of f correspond to n etremum of g? Eplin (e) Consider the function t ht f d Use grphing utilit to grph h Wht is the reltionship between g nd h? Verif our conjecture Find lim b evluting n pproprite n n sinin i n definite integrl over the intervl, () Show tht d d (b) Show tht b d b d () Show tht sin d cos d (b) Show tht sin n d cos n d, where n is positive integer Thermostt setting: Time (in hours) t True or Flse? In Eercises, determine whether the sttement is true or flse If it is flse, eplin wh or give n emple tht shows it is flse d C d C b b c d d sin d b sin d sin cos d cos C sin cos d sin C Assume tht f is continuous everwhere nd tht c is constnt Show tht cb b f d c f c d c () Verif tht sin u u cos u C u sin u du (b) Use prt () to show tht sin d Complete the proof of Theorem Show tht if f is continuous on the entire rel number line, then b bh f h d f d h b d d Putnm Em Chllenge If,,, re rel numbers stisfing n n show tht the eqution n n hs t lest one rel zero 6 Find ll the continuous positive functions f, for, such tht where f d f d f d n is rel number These problems were composed b the Committee on the Putnm Prize Competition The Mthemticl Assocition of Americ All rights reserved
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