Primitive of Power of Cosine of a x over Power of Sine of a x/Reduction of Both Powers

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Theorem

$\ds \int \frac {\cos^m a x} {\sin^n a x} \rd x = \frac {-\cos^{m - 1} a x} {a \paren {n - 1} \sin^{n - 1} a x} - \frac {m - 1} {n - 1} \int \frac {\cos^{m - 2} a x} {\sin^{n - 2} a x} \rd x + C$


Proof

With a view to expressing the primitive in the form:

$\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

let:

\(\ds u\) \(=\) \(\ds \cos^{m - 1} a x\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds -\paren {m - 1} a \cos^{m - 2} a x \sin a x\) Derivative of $\cos a x$, Derivative of Power, Chain Rule for Derivatives


and let:

\(\ds \frac {\d v} {\d x}\) \(=\) \(\ds \frac {\cos a x} {\sin^n a x}\)
\(\ds \) \(=\) \(\ds \sin^{-n} a x \cos a x\)
\(\ds \leadsto \ \ \) \(\ds v\) \(=\) \(\ds \frac {\sin^{-n + 1} a x} {a \paren {-n + 1} }\) Primitive of $\sin^n a x \cos a x$
\(\ds \) \(=\) \(\ds \frac {-1} {a \paren {n - 1} \sin^{n - 1} a x}\)


Then:

\(\ds \int \frac {\cos^m a x} {\sin^n a x} \rd x\) \(=\) \(\ds \int \cos^{m - 1} a x \frac {\cos a x} {\sin^n a x} \rd x\)
\(\ds \) \(=\) \(\ds \paren {\cos^{m - 1} a x} \paren {\frac {-1} {a \paren {n - 1} \sin^{n - 1} a x} }\) Integration by Parts
\(\ds \) \(\) \(\, \ds - \, \) \(\ds \int \paren {\frac {-1} {a \paren {n - 1} \sin^{n - 1} a x} } \paren {-\paren {m - 1} a \cos^{m - 2} a x \sin a x } \rd x + C\)
\(\ds \) \(=\) \(\ds \frac {-\cos^{m - 1} a x} {a \paren {n - 1} \sin^{n - 1} a x} - \frac {m - 1} {n - 1} \int \frac {\cos^{m - 2} a x} {\sin^{n - 2} a x} \rd x + C\) simplifying

$\blacksquare$


Also see


Sources

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