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

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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

$\blacksquare$


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