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Unformatted text preview: MTH 122 Calculus II Essex County College Division of Mathematics and Physics 1 Lecture Notes #3 Sakai Web Project Material 1 Integration by Parts Lets start with an example where were asked to verify: Z xe x d x = xe x- e x + C. As you know, we basically just have to check that: d d x ( xe x- e x + C ) = xe x . Here goes: d d x ( xe x- e x + C ) = d d x ( xe x )- d d x ( e x ) + d d x ( C ) = ( xe x + e x )- ( e x ) + (0) = xe x The main problem here is not the checking, but it is the actual process of finding the antideriva- tive. So far weve only used one real method, that was u-substitution, and although it is an important method it would fail miserably in this particular case. Another technique known as integration by parts uses the product rule . As you recall, the product rule is: d d x ( uv ) = u v + uv , where u and v are functions of x and u and v are the derivatives with respect to x . We can rewrite this product rule as follows: uv = d d x ( uv )- u v, and then integrate both sides,...
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