Here’s a neat substitution for integration that I never learned from calculus subjects but found in a book. I also frequently forget the formulae so whenever I want to use it I need to re-derive everything, hopefully writing it here will help me remember.

If is our integration variable, then we make the substitution . Using the trig identities, we can now get expressions for trigonometric functions in terms of .

.

And of course so .

Now we can work out things like , which is given by