# Primitive of Fourth Power of Sine of a x

## Theorem

$\displaystyle \int \sin^4 a x \ \mathrm d x = \frac {3 x} 8 - \frac {\sin 2 a x} {4 a} + \frac {\sin 4 a x} {32 a} + C$

## Proof

 $\displaystyle \int \sin^4 a x \ \mathrm d x$ $=$ $\displaystyle \int \left({\frac {3 - 4 \cos 2 a x + \cos 4 a x} 8}\right) \ \mathrm d x$ Power Reduction Formula for $\sin^4$ $\displaystyle$ $=$ $\displaystyle \frac 3 8 \int \mathrm d x - \frac 1 2 \int \cos 2 a x \ \mathrm d x + \frac 1 8 \int \cos 4 a x \ \mathrm d x$ Linear Combination of Integrals $\displaystyle$ $=$ $\displaystyle \frac {3 x} 8 - \frac 1 2 \int \cos 2 a x \ \mathrm d x + \frac 1 8 \int \cos 4 a x \ \mathrm d x + C$ Primitive of Constant $\displaystyle$ $=$ $\displaystyle \frac {3 x} 8 - \frac 1 2 \left({\frac {\sin 2 a x} {2 a} }\right) + \frac 1 8 \left({\frac {\sin 4 a x} {4 a} }\right) + C$ Primitive of $\cos a x$ $\displaystyle$ $=$ $\displaystyle \frac {3 x} 8 - \frac {\sin 2 a x} {4 a} + \frac {\sin 4 a x} {32 a} + C$ simplifying

$\blacksquare$