07_01 Integration by Parts.pdf - Math 1132Q 7.1 Integration by Parts As substitution provided a method to reverse the chain rule Integration by Parts

07_01 Integration by Parts.pdf - Math 1132Q 7.1 Integration...

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Math 1132Q 7.1: Integration by Parts As substitution provided a method to reverse the chain rule, Integration by Parts provides a means to reverse the product rule. Recall: ( ) ( ) ( ) udv vdu uv x g v x f u + = = = . Integrating, we get: ( ) + = = udv vdu uv uv , which simplifies to: - = vdu uv udv or ( ) ( ) ( ) ( ) ( ) ( ) dx x f x g x g x f dx x g x f + = . Example : x x x xe e xe dx d 3 3 3 3 3 + = implies dx e xe dx xe x x x - = 3 3 3 3 3 Approach : To solve dx xe x 3 , define dx e dv x u x 3 = = so 3 3 3 x x e dx e v dx du = = = , c e xe dx e xe vdu uv dx xe x x x x x + - = - = - = 9 3 3 3 3 3 3 3 3 Technique : Use integration by parts to simplify the integral. Select u and dv so that vdu is easier to solve than udv . Class work: 1. xdx x 3 cos 2. xdx x sin 2 3. dt e t t 2 4. xdx e x sin 5. dx e x x 2 3 6. dx x e 1 ln 7. dx x - 1 1 tan 8. xdx 5 sin

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