Section 8.1 Basic Integration Rules Fitting Integrands to Basic Rules

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.