Primitive of Cube of Hyperbolic Cotangent of a x/Proof 1
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Theorem
- $\ds \int \coth^3 a x \rd x = \frac {\ln \size {\sinh a x} } a - \frac {\coth^2 a x} {2 a} + C$
Proof
\(\ds \int \coth^3 a x \rd x\) | \(=\) | \(\ds \int \coth a x \coth^2 a x \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \coth a x \paren {1 + \csch^2 a x} \rd x\) | Difference of Squares of Hyperbolic Cotangent and Cosecant | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \coth a x \rd x + \int \coth a x \csch^2 a x \rd x\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\ln \size {\sinh a x} } a + \int \coth a x \csch^2 a x \rd x\) | Primitive of $\tanh a x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\ln \size {\sinh a x} } a - \frac {\coth^2 a x} {2 a} + C\) | Primitive of $\coth^n a x \csch^2 a x$: $n = 1$ |
$\blacksquare$