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

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Theorem

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


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:

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


and let:

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


Then:

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

$\blacksquare$


Also see


Sources