# Primitive of Cosine of a x over p plus q of Sine of a x

## Theorem

$\displaystyle \int \frac {\cos a x \rd x} {p + q \sin a x} = \frac 1 {a q} \ln \left\vert{p + q \sin a x}\right\vert + C$

## Proof

 $\displaystyle \frac \d {\d x} \left({p + q \sin a x}\right)$ $=$ $\displaystyle a q \cos a x$ Derivative of $\sin a x$ $\displaystyle \implies \ \$ $\displaystyle \int \frac {\cos a x \rd x} {p + q \sin a x}$ $=$ $\displaystyle \frac 1 {a q} \ln \left\vert{p + q \sin a x}\right\vert + C$ Primitive of Function under its Derivative

$\blacksquare$