# How to Integrate Odd Powers of Tangents with Secants

You can integrate odd powers of tangents with secants. To integrate tan^{m}*x* sec^{n}*x* when *m* is odd — for example, tan^{7} *x* sec^{9} *x* — you would follow these steps:

Peel off a tan

*x*and a sec*x*and place them next to the*dx**:*Use the trig identity tan

^{2}*x*= sec^{2}*x*– 1 to express the remaining tangent factors in terms of secants: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.