Integral of a power of a secant:

Note how in the last form we have grouped one power of the secant with the tangent. The reason becomes clear below.

If the correct factors are present, this can be handled by the Power rule for integrals:

, where and using the derivative of the secant, . Notice how one of the original powers of the secant is needed in u'dx.

Hence a combination of factors in the original problem can be replaced:

With this substitution, we get

Finally, we substitute for u to get the result in terms of x:

Notice that the argument of the resulting trig function is the same as that in the original problem.