Integration by Parts. Derive the following functions: y  xn y  sin x y  cos x y  tan x y  sec x y  csc x y  cot x y  ex y  ln x y  u ( x)  v( x) u ( x) 11. y  v( x) 12. y  u (v( x)) 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Integrate the following functions

y  xn y  sin x y  cos x y  ex 1 17. y  x There is a technique for integrating some products: Integration by parts (Manipulation of product rule) 13. 14. 15. 16.

dv

du

 u dxdx  uv   v dx dx or  udv  uv   vdu The integral of the product has to be written in the form:



(function)(differential of another function)

dv

 u dx dx or  udv Criterion 1: dv must be something you can integrate. Criterion #2: u should, if possible, be something that gets simpler when derived. Example 1:

 xe dx x

Example 2:  ln xdx

Example 3:

x

2

sin(3x)dx

Example 4:  e 2 x cos 3xdx

Example 5:

x

5

ln(4 x)dx