Simplify the expression within the integral. It is often possible to perform algebraic manipulation or substitution to remove complexity from the integral. This includes performing fraction simplifications, making trigonometric substitutions and changing terms to their inverses.
Attempt a simple "u - substitution" when possible. A vast number of integrals can be solved by simply performing an introductory level u - substitution. Even when a u - substitution fails to yield a final answer, it often provides important information concerning the dynamics of the integral itself.
Examine the integral and determine if it falls into a known form. Many integrals, especially those presented in classes, fall into a form that lends itself to an integration technique. (See Tips for more information.)
Evaluate the integral using the standard integration technique applicable to its classification, such as integration by parts or partial fraction decomposition.