How to Integrate Tangent/Secant Problems with an Odd, Positive Power of Tangent

Here’s how you integrate a trig integral that contains tangents and secants where the tangent power is odd and positive. You’ll need the tangent-secant version of the Pythagorean identity:

1. Lop off a secant-tangent factor and move it to the right.

   First, rewrite the problem:

   

2. Convert the remaining (even) tangents to secants with the tangent-secant version of the Pythagorean identity.

   

   Now make the switch.

   

3. Solve with the substitution method with u = sec(x) and du = sec(x) tan(x)dx.