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

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Theorem

$\displaystyle \int \frac {\sin a x \rd x} {p + q \cos a x} = \frac {-1} {a q} \ln \size {p + q \cos a x} + C$


Proof

\(\displaystyle \map {\frac \d {\d x} } {p + q \cos a x}\) \(=\) \(\displaystyle -a q \sin a x\) Derivative of $\cos a x$
\(\displaystyle \leadsto \ \ \) \(\displaystyle \int \frac {\sin a x \rd x} {p + q \cos a x}\) \(=\) \(\displaystyle \frac {-1} {a q} \ln \size {p + q \cos a x} + C\) Primitive of Function under its Derivative

$\blacksquare$


Also see


Sources