# How to know when to use integration by substitution vs. integration by parts?

Apr 8, 2015

Some problem types we learn to recognize:
$\int {\sin}^{n} x \cos x \mathrm{dx}$ Substitution.
$\int {e}^{a x} \mathrm{dx}$ substitution (or memorize)
$\int x {e}^{a x} \mathrm{dx}$ parts
$\int {x}^{n} \ln x \mathrm{dx}$ Parts if $n \ne - 1$

I'll describe a process, but it really amounts to:
try substitution fisrt, if that won't get you an answer, try parts. If that won't work either, try some other technique.

Look at the integrand to see if you can think of it as a product with one factor is the derivative or 'almost' the derivative of the other. That's a good suggestion for substitution.
Sometimes you'll need to regroup to make this work. Other times you may need multiple substitution.
Look at the integrand to see if you can think of it as a product with one factor you can differentiate and the other you can integrate. That's a good suggestion for parts.

Some integrals can be evaluated by either method. Most will not submit to either. (On an exam, most can be done by some method you've learned.)