Primitive of Cube of Cotangent of a x
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Theorem
- $\ds \int \cot^3 a x \rd x = \frac {-\cot^2 a x} {2 a} - \frac 1 a \ln \size {\sin a x} + C$
Proof
\(\ds \int \cot^3 x \rd x\) | \(=\) | \(\ds \int \cot a x \cot^2 a x \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \cot a x \paren {\csc^2 a x - 1} \rd x\) | Difference of Squares of Cosecant and Cotangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \cot a x \csc^2 a x \rd x - \int \cot a x \rd x\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-\cot^2 a x} {2 a} - \int \cot a x \rd x + C\) | Primitive of $\cot^n a x \csc^2 a x$: $n = 1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-\cot^2 a x} {2 a} - \paren {\frac {\ln \size {\sin a x} } a} + C\) | Primitive of $\cot a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-\cot^2 a x} {2 a} - \frac 1 a \ln \size {\sin a x} + C\) | simplifying |
$\blacksquare$
Also see
- Primitive of $\sin^3 a x$
- Primitive of $\cos^3 a x$
- Primitive of $\tan^3 a x$
- Primitive of $\sec^3 a x$
- Primitive of $\csc^3 a x$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\cot a x$: $14.442$