# Primitive of Power of Cosine of a x by Sine of a x

## Theorem

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

## Proof

 $\displaystyle z$ $=$ $\displaystyle \cos a x$ Simpson's Formula for Sine by Cosine $\displaystyle \leadsto \ \$ $\displaystyle \frac {\d z} {\d x}$ $=$ $\displaystyle -a \sin a x$ Primitive of $\cos a x$ $\displaystyle \leadsto \ \$ $\displaystyle \int \cos^n a x \sin a x \rd x$ $=$ $\displaystyle \int \frac {-z^n \rd x} a$ Integration by Substitution $\displaystyle$ $=$ $\displaystyle \frac {-z^{n + 1} } {\paren {n + 1} a} + C$ Primitive of Power $\displaystyle$ $=$ $\displaystyle \frac {-\cos^{n + 1} a x} {\paren {n + 1} a} + C$ substituting for $z$

$\blacksquare$