Primitive of x by Square of Hyperbolic Cotangent of a x
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Contents
Theorem
- $\displaystyle \int x \coth^2 a x \ \mathrm d x = \frac {x^2} 2 - \frac {x \coth a x} a + \frac 1 {a^2} \ln \left\vert{\sinh a x}\right\vert + C$
Proof
With a view to expressing the primitive in the form:
- $\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \ \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \ \mathrm d x$
let:
\(\displaystyle u\) | \(=\) | \(\displaystyle x\) | |||||||||||
\(\displaystyle \implies \ \ \) | \(\displaystyle \frac {\mathrm d u}{\mathrm d x}\) | \(=\) | \(\displaystyle 1\) | Derivative of Identity Function |
and let:
\(\displaystyle \frac {\mathrm d v}{\mathrm d x}\) | \(=\) | \(\displaystyle \coth^2 a x\) | |||||||||||
\(\displaystyle \implies \ \ \) | \(\displaystyle v\) | \(=\) | \(\displaystyle x - \frac {\coth a x} a\) | Primitive of $\coth^2 a x$ |
Then:
\(\displaystyle \int x \coth^2 a x \ \mathrm d x\) | \(=\) | \(\displaystyle x \left({x - \frac {\coth a x} a}\right) - \int \left({x - \frac {\coth a x} a}\right) \times 1 \ \mathrm d x + C\) | Integration by Parts | ||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle x^2 - \frac {x \coth a x} a + \int x \ \mathrm d x + \frac 1 a \int \coth a x \ \mathrm d x + C\) | Linear Combination of Integrals | ||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle x^2 - \frac {x \coth a x} a + \frac {x^2} 2 + \frac 1 a \int \coth a x \ \mathrm d x + C\) | Primitive of Power | ||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle x^2 - \frac {x \coth a x} a + \frac {x^2} 2 + \frac 1 a \frac {\ln \left\vert{\sinh a x}\right\vert} a + C\) | Primitive of $\coth a x$ | ||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \frac {x^2} 2 - \frac {x \coth a x} a + \frac 1 {a^2} \ln \left\vert{\sinh a x}\right\vert + C\) | simplifying |
$\blacksquare$
Also see
- Primitive of $x \sinh^2 a x$
- Primitive of $x \cosh^2 a x$
- Primitive of $x \tanh^2 a x$
- Primitive of $x \operatorname{sech}^2 a x$
- Primitive of $x \operatorname{csch}^2 a x$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\coth a x$: $14.622$