Primitive of x by Square of Hyperbolic Cosecant of a x
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Contents
Theorem
- $\displaystyle \int x \csch^2 a x \rd x = \frac {-x \coth a x} a + \frac 1 {a^2} \ln \size {\sinh a x} + C$
Proof
With a view to expressing the primitive in the form:
- $\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
\(\displaystyle u\) | \(=\) | \(\displaystyle x\) | |||||||||||
\(\displaystyle \leadsto \ \ \) | \(\displaystyle \frac {\d u} {\d x}\) | \(=\) | \(\displaystyle 1\) | Derivative of Identity Function |
and let:
\(\displaystyle \frac {\d v} {\d x}\) | \(=\) | \(\displaystyle \csch^2 a x\) | |||||||||||
\(\displaystyle \leadsto \ \ \) | \(\displaystyle v\) | \(=\) | \(\displaystyle -\frac {\coth a x} a\) | Primitive of $\csch^2 a x$ |
Then:
\(\displaystyle \int x \csch^2 a x \rd x\) | \(=\) | \(\displaystyle x \paren {-\frac {\coth a x} a} - \int \paren {-\frac {\coth a x} a} \times 1 \rd x + C\) | Integration by Parts | ||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \frac {-x \coth a x} a + \frac 1 a \int \coth a x \rd x + C\) | Linear Combination of Integrals | ||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \frac {-x \coth a x} a + \frac 1 a \frac {\ln \size {\sinh a x} } a + C\) | Primitive of $\coth a x$ | ||||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \frac {-x \coth a x} a + \frac 1 {a^2} \ln \size {\sinh a x} + C\) | 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 $x \coth^2 a x$
- Primitive of $x \sech^2 a x$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\csch a x$: $14.642$