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

Theorem

$\ds \int \frac {\cos a x \rd x} {\paren {p + q \sin a x}^n} = \frac {-1} {a q \paren {n - 1} \paren {p + q \sin a x}^{n - 1} } + C$

Proof

\(\ds z\)

\(=\)

\(\ds p + q \sin a x\)

\(\ds \leadsto \ \ \)

\(\ds \frac {\d z} {\d x}\)

\(=\)

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

Derivative of $\sin a x$

\(\ds \leadsto \ \ \)

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

\(=\)

\(\ds \int \frac {\d z} {a q z^n}\)

Integration by Substitution

\(\ds \)

\(=\)

\(\ds \frac 1 {a q} \int z^{-n} \rd z\)

Primitive of Constant Multiple of Function

\(\ds \)

\(=\)

\(\ds \frac 1 {a q} \frac {z^{-n + 1} } {-n + 1} + C\)

Primitive of Power

\(\ds \)

\(=\)

\(\ds \frac {-1} {a q \paren {n - 1} z^{n - 1} } + C\)

simplifying

\(\ds \)

\(=\)

\(\ds \frac {-1} {a q \paren {n - 1} \paren {p + q \sin a x}^{n - 1} } + C\)

substituting for $z$

$\blacksquare$

Also see

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

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.418$