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

Theorem

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

Proof

\(\ds \map {\frac \d {\d x} } {p + q \cos a x}\)

\(=\)

\(\ds -a q \sin a x\)

Derivative of $\cos a x$

\(\ds \leadsto \ \ \)

\(\ds \int \frac {\sin a x \rd x} {p + q \cos a x}\)

\(=\)

\(\ds \frac {-1} {a q} \ln \size {p + q \cos a x} + C\)

Primitive of Function under its Derivative

$\blacksquare$

Also see

• Primitive of $\dfrac {\cos a x} {p + q \sin a x}$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.415$