You can integrate even powers of secants with tangents. If you wanted to integrate tanm x secn x when n is even — for example, tan8 x sec6 x — you would follow these steps:

  1. Peel off a sec2 x and place it next to the dx:

  2. Use the trig identity 1 + tan2 x = sec2 x 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.