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

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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 $z$

$\blacksquare$


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