How to Integrate Odd Powers of Tangents with Secants

You can integrate odd powers of tangents with secants. To integrate tan^m x secⁿ x when m is odd — for example, tan⁷ x sec⁹ x — you would follow these steps:

1. Peel off a tan x and a sec x and place them next to the dx:

   

2. Use the trig identity tan²x = sec²x – 1 to express the remaining tangent factors in terms of secants:

   

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

   

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