Section 8.1 Basic Integration Rules Fitting Integrands to Basic Rules

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332460_0801.qxd 11/2/04 3:04 PM Page 521SECTION 8.1 Basic Integration Rules 521Trigonometric identities can often be used to fit integrals to one of the basicintegration rules.EXAMPLE 6Using Trigonometric IdentitiesFind tan 2 2x dx.TECHNOLOGY If you haveaccess to a computer algebra system,try using it to evaluate the integrals inthis section. Compare the form of theantiderivative given by the softwarewith the form obtained by hand.Sometimes the forms will be the same,but often they will differ. For instance,why is the antiderivative ln 2x Cequivalent to the antiderivativeln x C?Solution Note that tan 2 u is not in the list of basic integration rules. However, sec 2 uis in the list. This suggests the trigonometric identity tan 2 u sec 2 u 1. If you letu 2x, then du 2 dx and tan 2 2x dx 1 2 tan 2 u du 1 2 sec 2 u 1 du 1 2 sec 2 u du 1 2 du 1 2 tan u u 2 CSubstitution: u 2xTrigonometric identityRewrite as two integrals.Integrate. 1 tan 2x x C.2Rewrite as a function of x.This section concludes with a summary of the common procedures for fittingintegrands to the basic integration rules.Procedures for Fitting Integrands to Basic RulesTechniqueExpand (numerator).Separate numerator.Complete the square.Divide improper rational function.Add and subtract terms in numerator.Use trigonometric identities.Multiply and divide by Pythagorean conjugate.Example1 e x 2 1 2e x e 2x1 xx 2 1 1x 2 1 xx 2 112x x 12 1 x 1 2x 2x 2 1 1 1x 2 12xx 2 2x 1 2x 2 2x 2 2x 1 2x 2x 2 2x 1 2x 1 2cot 2 x csc 2 x 111 sin x 11 sin x 1 sin x1 sin x 1 sin x1 sin 2 x 1 sin xcos 2 x sec 2 x sin xcos 2 xNOTE Remember that you can separate numerators but not denominators. Watch out for thiscommon error when fitting integrands to basic rules.1x 2 1 1 x 2 1 1Do not separate denominators.
332460_0801.qxd 11/2/04 3:04 PM Page 522522 CHAPTER 8 Integration Techniques, L’Hôpital’s Rule, and Improper IntegralsExercises for Section 8.1In Exercises 1–4, select the correct antiderivative.1.2.3.dydx xx 2 1(a) 2x 2 1 C1(c) 2 x2 1 Cdydx xx 2 1(a) lnx2 1 C(c) arctan x Cdydx 1x 2 1(a) lnx2 1 C(c) arctan x Cdy4.dx x cosx2 1(a) 2x sinx 2 1 C1(c) 2 sinx2 1 C(b) x 2 1 C(d) lnx 2 1 C(b)2xx 2 1 C 2(d) lnx 2 1 C(b)2xx 2 1 C 2(d) lnx 2 1 C(b) 1 2 sinx2 1 C(d) 2x sinx 2 1 CIn Exercises 5–14, select the basic integration formula youcan use to find the integral, and identify u and a whenappropriate.2t 15. 3x 2 6. dxt27. 8. t 2 dt1x1 2x dx2t 12x9. 10. 4 dt31 t dt x11. t sin t 12. 4 dx2 dtsec 3x tan 3x dx13. cos xesin xdx114. xx 2 4 dxIn Exercises 15–50, find the indefinite integral.215. 16. 6x5 4 dxt 9 dt17. 2 518. t 2 t 3 3 1 dtz 4 dz19. 20. x 3 5 1 v 2x 32 dx3v 13 dvt x 121. 22. 2 3t xx 2x23. 24. 2 2x 4 dx3 9t 1 dt2x 1 dxx 4 dxe25. 26. 1x3x 1 11 e dx 3x 1 dxxSee www.CalcChat.com for worked-out solutions to odd-numbered exercises.27. 1 2x 28. x 1 1 x 2 2 dx3 dx29. x cos 2x 30. 2 dxsec 4x dxsin x31. csc x cot x dx 32. cos x dx33. e 34. 5x dxcsc 2 xe cot x dx235. 36. 5e 3e x 2 dxx 1 dxln x37. 2dx38. tan xlncos x dxx 1 sin x1 cos 39. dx40.cos xsin 1241. 42.cos 1 d3sec x 1 dx1143. 44. 1 2t 1 dt 4 3x dx45. 2 tan2t dt46. 2 e1tt t dt2 247.48.49.50.Slope Fields In Exercises 51–54, a differential equation, apoint, and a slope field are given. (a) Sketch two approximatesolutions of the differential equation on the slope field, one ofwhich passes through the given point. (b) Use integration tofind the particular solution of the differential equation and usea graphing utility to graph the solution. Compare the resultwith the sketches in part (a). To print an enlarged copy of thegraph, go to the website www.mathgraphs.com.dsdy51. 52.dx tan2 2xdt t1 t 4−136x x dx 21x 14x 2 8x 3 dx44x 2 4x 65 dx11 4x x dx 2 0, 1 21−1s1t0, 01−1 1−1ydx
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Section 8.1 Basic Integration Rules Fitting Integrands to Basic Rules

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