Primitive of x by Hyperbolic Sine of a x

Theorem

 * $\ds \int x \sinh a x \rd x = \frac {x \cosh a x} a - \frac {\sinh a x} {a^2} + C$

Proof
With a view to expressing the primitive in the form:
 * $\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

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 $x \sech a x$
 * Primitive of $x \csch a x$