How do you know which method of integration to use?

It takes doing a lot of problems to be able to see which to use just by glancing at it but here's a nice little check list to go through.
1. Does substitution work?  (ALWAYS CHECK THIS ONE FIRST)
2. Is it a basic integral?
3. Is it a ratio of polynomials?
4. Is it a product of trig functions?
5. Is it a product of two different functions?
6. Can you make a trigonometric substitution?

After that it gets a little hairy because it might require multiple choices of these.  If you can make a substitution go for that because it's the easiest to work with.  Being a basic integral is kind of obvious, power rule, trig, etc... Ratios of polynomials typically require substitution, division, partial fractions, or trig sub.  Product of trig functions is trig integrals, you'll get to that.  Two different functions is where by parts comes in.  Stuff like a polynomial times a trig functions or an exponential and a trig, etc... And you'll get to trig sub.

Keep practicing and good luck!