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Section5_5_review - Section 5.5 The Substitution Rule gx)x...

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Section 5.5: The Substitution Rule For indefinite integrals of the form fgx ( 29 ( 29 g x ( 29 dx , let u = gx (29 . Then du =′ g x ( 29 dx , and the original integral can be expressed as f u (29 du . Choose u = gx (29 such that its derivative also appears in the original integral, except perhaps for a constant. In other words, follow these steps: (1) make the substitution u = gx (29 ; (2) determine du =′ g x ( 29 dx ; (3) write the integral in terms of u ; (4) integrate in terms of u ; (5) back- substitute to return to variable x . For definite integrals of the form f gx (29 ( 29 g x (29 a b dx , again let u = gx (29 . Then du =′ g x ( 29 dx , u lower = ga ( 29 , u upper = gb ( 29 , and the original integral can be expressed as f u ( 29 u lower u upper du . In other words, follow these steps: (1) make the substitution u = gx (29 ; (2) determine du =′ g x ( 29 dx ; (3) change the limits of integration to be valid for u , where u lower = ga ( 29 and u upper = gb ( 29 ; (4) write the integral in terms of u ; (5) integrate in terms of u ; (6) evaluate at the limits of integration. Example. Evaluate r r 2 + b 2 ( 29 3 2 dr , where b is a constant. Solution. Let u = r 2 + b 2 Then du = 2 rdr So rdr = du 2 So r r 2 + b 2
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