Integration via a Change of Variables

Transcription

1 LECTURE 33 Integration via a Change of Variables In Lectures an 3 where we evelope a general technique for computing erivatives that was base on two ifferent kins of results. First, we ha a table that liste the erivatives of a relatively small set of elementary functions x xn =nx n x eλx =λe λx x ln x = x x ax =ln a a x x sin(x)=cos(x) x cos(x)= sin(x) x tan(x)=sec (x) x cot(x)= csc (x) x sin (x)= x x cos (x)= x x tan (x)= +x x cot (x)= +x Seconly, we ha a short list of rules by which the erivatives of more complicate functions coul be reuce to the computations involving the erivatives liste above: f TheSumRule: (f+g)= x x +g x The Constant Multiplier Rule: x (cf)=cf x ifcisaconstant f The Prouct Rule: (fg)= x ( x g+fg x f The Quotient Rule: )= f g fg x g g f TheChainRule: (f(g(x)))= g x u u=g(x) x,wheref := f x,g := g x Similarly, we have evelope rules for computing the integrals (a.k.a. anti-erivatives) of functions by knowing a few basic anti-erivatives an a few rules for computing the anti-erivatives of more complicate functions built from our set of basic function. In fact, our Integral Table can be obtaine irectly from the Table of Derivatives by reinterpreing the function being ifferentiate as the anti-erivative of its erivative: x n x= n+ xn+ +C cos(x)x=sin(x)+c x x=sin (x)+c e λx x= λ eλx +C sin(x)= cos(x)+c +x x=tan (x)+c x x=ln x +C sec (x)x=tan(x)+c a x x= ax ln a +C csc (x)x= cot(x)+c However, our corresponing list of Integration Rules(at least right now) is consierably shorter: TheSumRule: (f(x)+g(x))x= f(x)x+ g(x)x The Constant Multiplier Rule: cf(x)x=c f(x)x Toay,weshallaanewruleofintegration,theChangeofVariableRule,whichisbasicallytheanalogof the Chain Rule for ifferentiation. (This Change of Variable Rule is also calle the Substitution Rule.) 40

2 33. INTEGRATION VIA A CHANGE OF VARIABLES 4 Theorem33.(TheChangeofVariablesRule). Supposetheintegranofanintegralisofthetheform g(h(x))h (x) (with g(x) continuous an h(x) ifferentiable), then u=h(x) () g(h(x))h (x)x= g(u)u+c Proof. Thisisveryeasy. Tocompute g(h(x))h (x)x weneetofinananti-erivativeofg(h(x))h (x). SupposeGisananti-erivativeofg(x);then g(u)u G(x)+C an () u=h(x) g(u)u=g(h(x))+c But, now if we ifferentiate the right-han-sie of(), applying the Chain Rule, we get G (h(x))h (x) g(h(x))h (x) Thisshowsthat u=h(x) g(u)uisananti-erivativeofg(h(x))h (x). Thus g(h(x))h (x)x=anti-erivativeofg(h(x))h (x)= u=h(x) g(u)u Example 33.. Compute using the Change of Variables Rule. xsin ( x + ) x Thekeythinghereistofigureouthowtoviewtheintegran sin ( x + ) x asbeingoftheformg(h(x))h (x). Supposewetakeh(x)=x +ang(u)=sin(u). Then g(h(x))h (x)=sin ( x + ) (x) which is exactly our integran. We can now apply the Change of Variables Theorem xsin ( x + ) u=x + x= sin(u)u+c= cos(u) u=x + +C= cos ( x + ) +C Remark33.3. A convenient way to remember the Change of Variable Rule is via a mneumonic arising fromthenotionofifferentials. Recallthatifuisafunctionofx,itsifferentialis u=u (x)x Ifu=h(x),thenmeans an u=h (x)x g(u)u= g(h(x))h (x)x.

3 33. INTEGRATION VIA A CHANGE OF VARIABLES 4 or (3) g(h(x))h (x)x= g(u)u which is how the text writes the Change of Variable Rule. It is important to remember though, that computetherighthansieofbyfirstfininganti-erivativeg(u)ofg(u),anthensubstituting h(x)for uinthefinalresultg(h(x)). (Presumably,thisiswhythetextcallsthismethotheSubstitutionRule). Example Compute xe x x Wewanttoviewtheintegranasbeingoftheformg(u)u. Ititnaturaltotry However, the integran is not quite g(u) = e u u(x) = x = u=xx g(u)u=xe x we reoffbyafactorof. Ontheotherhan,wecanwrite xe x x= xe x x= ( ) e x x x x= Example Try to compute e x x e u u= eu = ex Herewemighttrythesamething Again we have g(u) = e u u(x) = x = u=xx g(u)u=e x (x)x butwehaveamoreseriousproblem. Wecannotgetrioftheextrafactorof(x)bysimplymulitplying by a constant factor. In fact, the integral e x x isoneoftheclassicalexamplesofaintegralforwhichcannotbecomputeintermsofelementaryfunctions. (Itisperfectlywellefineasanintegral,it sjustthattheanswercan tbeexpresseintermsofelementary functions.) InthenextcoupleofexamplesI lltrytoemonstratethesortofthinkingprocessthatoneneestousein orer to successfully apply the Change of Variables Metho. Example Compute e sin(x+) cos(x+)x

4 33. INTEGRATION VIA A CHANGE OF VARIABLES 43 Thefirstthingtolookforisthemostcomplicatefactorintheintegran;moreprecisely,thefactor thatisthemostcomplicatecompositionoffunctions. Inthecaseathan,itisthefactor which is a composition where e sin(x+) x x+ sin(x+) e sin(x+). Thelastoperationapplieissin(x+) e sin(x+) anthissuggeststhatweshoultrythinking ofthisfactorase u withu=sin(x+). Now ( ) u=sin(x+) = u= x sin(x+) x=(cos(x+))x=cos(x+)x Letusthinkofthislastequationas x= cos(x+) u Insie the orginal integral we now make the substitutions e sin(x+) e u x cos(x+) u toget e sin(x+) cos(x+)x= e u cos(x+) cos(x+) u= e u u= eu +C Thefinalstepistorememberthatu=sin(x+). Aftermakingthislastsubstitutionweget e sin(x+) cos(x+)x= esin(x+) +C Example Compute x 4x x The integran factors as x 4x Theseconfactorisacompositionoftheform x 4x =( 4x ) / 4x which suggests that we try g(u)=u / sinceu u / corresponstothelastfunctionapplieanu= 4x,sincethatistheargument ofthelastfunctiontoapplie. Wehave or We now make the substitutions u= 4x = u= ( 4x ) x= 8xx 4x x= 8x u u / x 8x u

5 33. INTEGRATION VIA A CHANGE OF VARIABLES 44 into the original integran. This yiels ( x x xu / ) u= 4x 8x 8 = 8 (u /) u / u= 8 ( +u /+ ) +C = 4 u/ = 4 4x +C