Integration by parts
From AoPSWiki
The purpose of integration by parts is to replace a difficult integral with an easier one. The formula is
Order
Now, given an integrand, what should be 
and what should be 
? Since will show up as 
and as 
in the integral on the RHS, u should be chosen such that it has an "easy" (or "easier") derivative and so that it has a easy antiderivative.
A mnemonic for when to substitute for what is LIATE:
Logarithmic
Inverse trigonometric
Algebraic
Trigonometric
Exponential
If any two of these types of functions are in the function to be integrated, the type higher on the list should be substituted as u.
Examples
x has a pretty simple derivative, so let's say 
. Then 
, 
, and 
. We have
. You can take the derivative to see that it is indeed our desired result.
Compute 
.
