Primitive of Square of Hyperbolic Cosecant of a x over Hyperbolic Cotangent of a x
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Theorem
- $\ds \int \frac {\csch^2 a x \rd x} {\coth a x} = \frac {-1} a \ln \size {\coth a x} + C$
Proof
\(\ds \frac \d {\d x} \coth a x\) | \(=\) | \(\ds -a \csch^2 a x\) | Derivative of $\coth a x$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \frac {-a \csch^2 a x \rd x} {\coth a x}\) | \(=\) | \(\ds \ln \size {\coth a x} + C\) | Primitive of Function under its Derivative | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \frac {\csch^2 a x \rd x} {\coth a x}\) | \(=\) | \(\ds \frac {-1} a \ln \size {\coth a x} + C\) | Primitive of Constant Multiple of Function |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\coth a x$: $14.619$