5.5 U-substitution. Solution. Z

Transcription

1 CHAPTER 5. THE DEFINITE INTEGRAL U-sbstittion Eample. (a) Fin the erivative of sin( 2 ). (b) Fin the anti-erivative cos( 2 ). Soltion. (a) We se the chain rle: sin(2 )=cos( 2 )( 2 ) 0 =cos( 2 )2 (b) We gess an check, starting with or reslt from part (a):?=cos(2 ) sin(2 )=2cos( 2 ) No. Bt close. 2 sin(2 )=cos( 2 )X Writing or answer in the other irection we get cos( 2 )= 2 sin(2 )+C. Eample 2. (a) What is e? What is e? (b) Let = What is? (c) Solve for. () Trn e into a new integral by sbstitting with an with yor answer to part (c). (e) Fin (6 + 2)e (f) Doble check yor answer by taking the erivative. Soltion. (a) e = e + C, e = e + C. (b) =6 +2. (c) =(6 +2) () z } { z } { e = e (6 +2) = e (6 +2)

2 CHAPTER 5. THE DEFINITE INTEGRAL 23 (e) (6 +2)e = e = e + C = e C. (f) e C = e = e (6 +2)X Eample 3. (a) Fin the anti-erivative sin(3 + 7). (b) Fin the anti-erivative 7 sin( 8 ). (c) Fin the anti-erivative (e + )sin(2e + 2 ). Soltion. In the following soltion we (as sal) go to one etreme in showing many steps. With a little practice, abot half of the steps below will be nnecessary. With a lot of practice, almost all of the steps will be nneccessary. (a) =3 +7 =(3 +7) 0 =3 0 = 3 sin(3 +7) = sin(3 {z +7 } ) = sin() 3 {z} 3 = 3 ( cos()) + C = cos(3 +7) 3 (b) = 8 =( 8 ) 0 =8 7 7 = 8

3 CHAPTER 5. THE DEFINITE INTEGRAL 24 (c) 7 sin( 8 ) = sin( {z} 8 ) 7 {z } 8 = sin() 8 = sin() 8 = 8 cos()+c = 8 cos(8 )+C =2e + 2 =(2e +2) =2(e + ) (e + ) = 2 (e + )sin(2e + 2 ) = sin(2e {z + } 2 )(e + ) {z } 2 = sin() 2 = sin() 2 Eample 4. (b) Fin (a) Fin = 2 cos() = 2 cos(2e + 2 )+C Soltion. (a) To get an iea of what shol be, westartbylookingforwhatmight be, i.e. bylookingfor?,wherewecolhave? eqal the eriavtive of : = 3 + = {z 4 } goo? {z } goo? To ecie which option to se, ask if we col make eqal to something else in the integral, so that it s erivative is what we ve highlighte for.

4 CHAPTER 5. THE DEFINITE INTEGRAL 25 The secon option is better: to make 3 = we can pick = 4. = 4 =4 3 3 = = + 4 {z} 3 {z } 4 (b) Let s look at the options again + = 4 + = {z 4 } goo? + 4 {z} goo? This one is trickier. The key is to remember: we only nee to fin some part of the rest of the integral to call so that it s erivative shows p in. So,areyoevergoingtofinsomethingin to call, sothatits erivative is? Not likely. +4 Can yo fin something in to call, so that its erivative is +4? What wol have to be? It wol have to be 2.Istherean 2 in + 4?Yes:4 =( 2 ) 2. = 2 =2 + 4 = 2 Eample 5. Fin or last basic fnctions.) Soltion. tan(). (Note: this gives s the anti-erivative of one of We start by rewriting the fnction as a fraction: sin() cos(). The obvios choices for (an the reslting ) are =sin() =cos() =cos() = sin()

5 CHAPTER 5. THE DEFINITE INTEGRAL 26 sin() The original integral oes not have cos(), it has since cos() cos() = sin() cos(). Ths, we nee to se =cos() an = sin(). Now we shol look at the original integral an see if we can fin how to translate all the s. Here s how it looks ( ) sin() cos() {z }. If yo replace the inicate parts shown with an, yo shol get this sin() cos() =. Yo shol be able to oble check that we ve one or work right: if yo sbstitte back cos() inplaceof, an sin() in place of, thenyo get the original integral. Now we finish p: = ln + C = ln cos() + C A this to yor list of basic anti-erivative facts tan() = ln cos() + C =ln sec() + C cos(ln()) Eample 6. Fin Soltion. Yor intition shol be that can eqal what is insie of cosine. =ln() = z } { cos( ln() ) = cos() =sin()+c =sin(ln()) + C 7sin() cos() Eample 7. Fin + cos 2 () Soltion. Some gesses yo col make in solving this integral =sin() =+cos 2 ()

6 CHAPTER 5. THE DEFINITE INTEGRAL 27 =cos() =cos 2 () =sin()cos() =2sin()cos() Yo o not nee to instantly see which of these choices is best, bt yo wol figre it ot by looking at in each case (i.e. the erivative of what eqals). Then, yo wol ask yorself, if yo replace parts of the original integral with an, wolyogetanintegralthatyocanfinish? Inotherwors,wol f() be an integral that is on or short list of basic anti-erivatives (c.f. page??)? Hopeflly yo see that =+cos 2 () works. Here s how 7sin()cos() +cos 2 () 2 =+cos 2 () = 2sin()cos() =sin()cos() 2 7 = 2 = 7 2 ln = 7 2 ln( + cos2 ()) + C