Integration by parts
The purpose of integration by parts is to replace a difficult integral with an easier one. The formula is
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:
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.
x has a pretty simple derivative, so let's say . Then , , and . We have
, where is the constant of integration. You can take the derivative to see that it is indeed our desired result.