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
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.426$