Try to apply the fundamental theorem. If derivative (X) = Y, then integral(Y) = X. If you can find an expression whose derivative is the expression you want to take the integral of, you are done. This will be true in only the simplest cases, but it is the first thing to check.
Look at a table of integrals. A good calculus book will have these in an appendix. Often the most common forms are printed on the inside covers. If the function you want to integrate fits one of these forms, you are done.
Learn the techniques needed to extend the basic forms from the table of integrals. These techniques include integration by parts, by change of variables, and dozens more. All of these techniques are explained in a good calculus book.
A really easy way to calculate integrals, that almost always works, is to use one of the symbolic software packages. Two of these that are commonly available are Maple and Mathematica. If you have one of these, the calculation of most integrals is just a few keystrokes away.