Calculus II Notes 5.5 - Calculus II-Stewart Dr. Berg Spring...

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Calculus II- Stewart Dr. Berg Spring 2010 Page 1 5.5 5.5 The Substitution Rule The substitution rule reverses the chain rule. Recall that, by the chain rule, d dx f ( g ( x )) [ ] = f ( g ( x )) g ( x ) . Reversing this yields f ( g ( x )) g ( x ) dx = d dx f ( g ( x )) [ ] dx = f ( g ( x )) + C . We simplify this process by using substitution. Theorem If u = g ( x ) is differentiable and f is continuous on the range of g , then f ( g ( x )) g ( x ) dx = f ( u ) du . Note: We treat dx and du = g ( x ) dx as if they were differentials. Example A Find x 3 1 ( ) 7 3 x 2 dx . Solution : Let u = x 3 1 . Then du = 3 x 2 dx so that x 3 1 ( ) 7 3 x 2 dx = u 7 du = u 8 8 + C = x 3 1 ( ) 8 8 + C . Example B Find x 2 x 3 + 4 dx . Solution : Let u = x 3 + 4 . Then du = 3 x 2 dx so that x 2 x 3 + 4 dx = 1 3 x 3 + 4 ( ) 1/ 2 3 x 2 dx = 1 3 u 1/ 2 du = 1 3 u 3 / 2 3/2 + C = 2 9 u 3 / 2 + C = 2 9 x 3 + 4 ( ) 3 / 2 + C . An equivalent method is to solve for the differential of x . Indeed, if du = 3 x 2 dx then dx = du 3 x 2 , so that x 2 x 3 + 4 dx = u ( ) 1/ 2 x 2 du 3 x 2 = 1 3 u 1/ 2 du = 1 3 u 3 / 2 + C = 2 9 u 3 / 2 + C = 2 9 x 3 + 4 ( ) 3 / 2 + C .
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Calculus II- Stewart Dr. Berg Spring 2010 Page 2 5.5 Example C Find sin x cos x dx . We present three possible solutions. Solution : a) Let u = sin x . Then du = cos x dx so that sin x cos x dx = udu = u 2 2 + C = 1 2 sin 2 x + C .
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Calculus II Notes 5.5 - Calculus II-Stewart Dr. Berg Spring...

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