Primitive of Power of Cosine of a x over Power of Sine of a x/Reduction of Power of Sine
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Theorem
- $\ds \int \frac {\cos^m a x} {\sin^n a x} \rd x = \frac {-\cos^{m + 1} a x} {a \paren {n - 1} \sin^{n - 1} a x} - \frac {m - n + 2} {n - 1} \int \frac {\cos^m a x} {\sin^{n - 2} a x} \rd x + C$
Proof
\(\ds \) | \(\) | \(\ds \frac {-\cos^{m + 1} a x} {a \paren {n - 1} \sin^{n - 1} a x} - \frac {m + 1} {n - 1} \int \frac {\cos^m a x} {\sin^{n - 2} a x} \rd x + C\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\cos^{m + 2} a x} {\sin^n a x} \rd x\) | Primitive of $\dfrac {\cos^m a x} {\sin^n a x}$: Reduction of Both Powers | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\cos^m a x \paren {1 - \sin^2 a x} } {\sin^n a x} \rd x\) | Sum of Squares of Sine and Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\cos^m a x} {\sin^n a x} \rd x - \int \frac {\cos^m a x \sin^2 a x} {\sin^n a x} \rd x\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\cos^m a x} {\sin^n a x} \rd x - \int \frac {\cos^m a x} {\sin^{n - 2} a x} \rd x\) | simplifying |
Hence, rearranging:
\(\ds \int \frac {\cos^m a x} {\sin^n a x} \rd x\) | \(=\) | \(\ds \frac {-\cos^{m + 1} a x} {a \paren {n - 1} \sin^{n - 1} a x} - \frac {m + 1} {n - 1} \int \frac {\cos^m a x} {\sin^{n - 2} a x} \rd x + \int \frac {\cos^m a x} {\sin^{n - 2} a x} \rd x + C\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-\cos^{m + 1} a x} {a \paren {n - 1} \sin^{n - 1} a x} - \frac {m + 1} {n - 1} \int \frac {\cos^m a x} {\sin^{n - 2} a x} \rd x - \frac {-n + 1} {n - 1} \int \frac {\cos^m a x} {\sin^{n - 2} a x} \rd x + C\) | common denominator | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-\cos^{m + 1} a x} {a \paren {n - 1} \sin^{n - 1} a x} - \frac {m - n + 2} {n - 1} \int \frac {\cos^m a x} {\sin^{n - 2} a x} \rd x + C\) |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sin a x$ and $\cos a x$: $14.427$