Primitive of x by Hyperbolic Sine of a x

Theorem

 * $\displaystyle \int x \sinh a x \ \mathrm d x = \frac {x \cosh a x} a - \frac {\sinh a x} {a^2} + C$

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:

and let:

Then:

Also see

 * Primitive of $x \cosh a x$
 * Primitive of $x \tanh a x$
 * Primitive of $x \coth a x$
 * Primitive of $x \operatorname{sech} a x$
 * Primitive of $x \operatorname{csch} a x$