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

## Theorem

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

## Proof

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

$\blacksquare$