Primitive of Power of Secant of a x

Theorem

 * $\displaystyle \int \sec^n a x \rd x = \frac {\sec^{n - 2} a x \tan a x} {a \paren {n - 1} } + \frac {n - 2} {n - 1} \int \sec^{n - 2} a x \rd x$

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

let:

and let:

Then:

Also see

 * Primitive of $\sin^n a x$
 * Primitive of $\cos^n a x$
 * Primitive of $\tan^n a x$
 * Primitive of $\cot^n a x$
 * Primitive of $\csc^n a x$