How to Integrate Even Powers of Secants with Tangents

You can integrate even powers of secants with tangents. If you wanted to integrate tan^m x secⁿ x when n is even — for example, tan⁸x sec⁶x — you would follow these steps:

1. Peel off a sec²x and place it next to the dx:

   

2. Use the trig identity 1 + tan² x = sec²x to express the remaining secant factors in terms of tangents:

   

3. Use the variable substitution u = tan x and du = sec²x dx:

   

At this point, the integral is a polynomial, and you can evaluate it.