Primitive of Square of Hyperbolic Cotangent Function
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Theorem
- $\ds \int \coth^2 x \rd x = x - \coth x + C$
where $C$ is an arbitrary constant.
Proof
| \(\ds \int \coth^2 x \rd x\) | \(=\) | \(\ds \int \paren {1 + \csch^2 x} \rd x\) | Difference of Squares of Hyperbolic Cotangent and Cosecant | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int 1 \rd x + \int \csch^2 x \rd x\) | Linear Combination of Primitives | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int 1 \rd x - \coth x + C\) | Primitive of Square of Hyperbolic Cosecant Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds x - \coth x + C\) | Primitive of Constant |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: General Rules of Integration: $14.34$