How do you solve integral problems?

Answer 1

That depends on the specific problem. There is no simple procedure that will let you systematically solve any integral. There are even integrals that look fairly simple (such as the integral of (sin x) / x, if I remember correctly), which can't be solved as a finite number of elementary functions.

You need to learn different tricks and techniques (taught in calculus books), know when you can apply each one, and quite often, experiment.