Primitive of Square of Hyperbolic Cotangent of a x
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Theorem
- $\ds \int \coth^2 a x \rd x = x - \frac {\coth a x} a + C$
Proof
| \(\ds \int \coth^2 x \rd x\) | \(=\) | \(\ds x - \coth x + C\) | Primitive of $\coth^2 x$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \int \coth^2 a x \rd x\) | \(=\) | \(\ds \frac 1 a \paren {a x - \coth a x} + C\) | Primitive of Function of Constant Multiple | ||||||||||
| \(\ds \) | \(=\) | \(\ds x - \frac {\coth a x} a + C\) | simplifying |
$\blacksquare$
Also see
- Primitive of $\sinh^2 a x$
- Primitive of $\cosh^2 a x$
- Primitive of $\tanh^2 a x$
- Primitive of $\sech^2 a x$
- Primitive of $\csch^2 a x$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\coth a x$: $14.616$
- 1968: George B. Thomas, Jr.: Calculus and Analytic Geometry (4th ed.) ... (previous) ... (next): Back endpapers: A Brief Table of Integrals: $126$.