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

Theorem

 * $\displaystyle \int \sin^m a x \cos^n a x \rd x = \frac {-\sin^{m - 1} a x \cos^{n + 1} a x} {a \paren {m + n} } + \frac {m - 1} {m + n} \int \sin^{m - 2} a x \cos^n a x \rd x + C$

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:

Hence after rearranging:

Also see

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