# Primitive of Square of Cosine of a x

## Theorem

$\displaystyle \int \cos^2 a x \, \mathrm d x = \frac x 2 + \frac {\sin 2 a x} {4 a} + C$

## Proof

 $\displaystyle \int \cos^2 x \, \mathrm d x$ $=$ $\displaystyle \frac x 2 + \frac {\sin 2 x} 4 + C$ Primitive of $\cos^2 x$ $\displaystyle \implies \ \$ $\displaystyle \int \cos^2 a x \, \mathrm d x$ $=$ $\displaystyle \frac 1 a \left({\frac {a x} 2 + \frac {\sin 2 a x} 4}\right) + C$ Primitive of Function of Constant Multiple $\displaystyle$ $=$ $\displaystyle \frac x 2 + \frac {\sin 2 a x} {4 a} + C$ simplifying

$\blacksquare$