# Primitive of x by Square of Hyperbolic Cosecant of a x

## 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

$\blacksquare$