Problem: Evaluate the integral
Solution: This is a root of a quadratic expression, and we have a box exactly for this type. The procedure calls first for the completion of a square:
So we can integrate:
In that box we also mentioned an alternative method using Euler
substitutions. Since the expression under the root can be factored, we can
use the third substitution. So we factor the expression as
Having two real roots, we could also do the third substitution this way:
This particular problem could be also solved in another way, based on an algebraic trick. It is not standard, so it is hard to advise how one could come up with it.
Check that all three answers are correct.