It’s useful to know when you should avoid using trig substitution. With some integrals, it’s better to expand the problem into a polynomial. For example, look at the following integral:
This may look like a good place to use trig substitution, but it’s an even better place to use a little algebra to expand the problem into a polynomial:
Similarly, look at this integral:
You can use trig substitution to evaluate this integral if you want to. (You can also walk to the top of the Empire State Building instead of taking the elevator if that tickles your fancy.) However, the presence of that little x in the numerator should tip you off that variable substitution will work just as well:
Using this substitution results in the following integral:
Done!