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 