Primitive of Power of Secant of a x

Theorem

 * $\displaystyle \int \sec^n a x \ \mathrm d x = \frac{\sec^{n - 2} a x \tan a x} {a \left({n - 1}\right)} + \frac {n - 2} {n - 1} \int \sec^{n - 2} a x \ \mathrm d x$

Proof
With a view to expressing the primitive in the form:
 * $\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \ \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \ \mathrm d x$

let:

and let:

Then:

Also see

 * Primitive of $\csc^n a x$