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Integration by Parts Integration by Parts is based on the product rule for derivatives. If F and G are differentiable functions, then (FG)′(x) = F(x)G′(x)+F′(x)G(x) We can rearrange this to: F(x)G′(x) = (FG)′(x) – F′(x)G(x) then integrate with respect to x ∫ F(x)G′(x)dx = ∫(FG)′(x)dx – ∫F′(x)G(x)dx then ∫ F(x)G′(x)dx = (FG)(x) – ∫F′(x)G(x)dx If f(x) can be rewritten as the product of F(x)G′(x) then ∫f(x)dx = (FG)(x) – ∫F′(x)G(x)dx Let u = F(x) and dv = G′(x)dx then du = F′(x)dx and v = G(x) and f(x) = udv then ∫f(x)dx = ò u dv = uv – ∫vdu To use integration by parts, break the integrand into 2 parts: u and dv. The choice for u is based on LIPET Remember dv must be something that is easily integrated. Examples: 1. ∫xcosxdx 2. ∫xe-xdx 3. ∫ylnydy 4. ∫x2exdx Homework: 1. 2 xe x dx 2. x5 ln xdx 3. ln xdx 4. x sin xdx 5. x 2 cos xdx 6. x 2 5 x e x dx