# Thread: When to use each method of integration?

1. ## When to use each method of integration?

My problem for a while has been deciding which method of integration to use in every given situation.

Are there times when you could use either Integration by Parts or U-substitution to achieve the same answer? Is this always/never the case?

In what situations do you know for sure to use each of these methods?

2. ## Re: When to use each method of integration?

It's a matter of experience, and being able to "see" how to do it. I can think of integrals where multiple substitutions all work; it wouldn't surprise me if there are integrals that succumb to multiple methods. Here's an example:

$\displaystyle \int_{0}^{10}(-2x)e^{-x^{2}}\,dx.$

I could substitute either $\displaystyle u=e^{-x^{2}}$ or $\displaystyle u=-x^{2},$ and I'd get the same result either way, as I should.

3. ## Re: When to use each method of integration?

My usual way is to try the simplest thing I think will work and if it doesn't, try something else!

4. ## Re: When to use each method of integration?

You have a lot of integration techniques but in the first place it's important I think to look if you're integrand is:
- rational function (often: achieve the euclidian division, split in partial fractions, ...)
- a product of two functions (often: use integration by parts, ...)
- irrational function (often: goniometric substitution, ...)
- trigoniometric function (often: formulas of Simpson, using t-formulas, ... -> good acknowledge of goniometric formulas)
- ...

But offcourse the most important thing is that you can use substitution.