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

## Theorem

$\displaystyle \int \frac {\sin a x \ \mathrm d x} {\left({p + q \cos a x}\right)^n} = \frac 1 {a q \left({n - 1}\right) \left({p + q \cos a x}\right)^{n - 1} } + C$

## Proof

 $\displaystyle z$ $=$ $\displaystyle p + q \cos a x$ $\displaystyle \implies \ \$ $\displaystyle \frac {\mathrm d z} {\mathrm d x}$ $=$ $\displaystyle -a q \sin a x$ Derivative of $\cos a x$ $\displaystyle \implies \ \$ $\displaystyle \int \frac {\sin a x \ \mathrm d x} {\left({p + q \cos a x}\right)^n}$ $=$ $\displaystyle \int \frac {\mathrm d z} {- a q z^n}$ Integration by Substitution $\displaystyle$ $=$ $\displaystyle \frac {-1} {a q} \int z^{-n} \ \mathrm d z$ Primitive of Constant Multiple of Function $\displaystyle$ $=$ $\displaystyle \frac {-1} {a q} \frac {z^{-n+1} } {-n + 1} + C$ Primitive of Power $\displaystyle$ $=$ $\displaystyle \frac 1 {a q \left({n - 1}\right) z^{n - 1} } + C$ simplifying $\displaystyle$ $=$ $\displaystyle \frac 1 {a q \left({n - 1}\right) \left({p + q \cos a x}\right)^{n - 1} } + C$ substituting for $z$

$\blacksquare$