Primitive of Power of Sine of a x by Power of Cosine of a x/Reduction of Power of Cosine

Theorem

 * $\displaystyle \int \sin^m a x \cos^n a x \ \mathrm d x = \frac {\sin^{m + 1} a x \cos^{n - 1} a x} {a \left({m + n}\right)} + \frac {n - 1} {m + n} \int \sin^m a x \cos^{n - 2} a x \ \mathrm d x + C$

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:

Hence after rearranging:

Also see

 * Primitive of $\sin^m a x \cos^n a x$ : Reduction of Power of Sine