Primitive of Power of x by Hyperbolic Sine of a x

Theorem

 * $\displaystyle \int x^m \sinh a x \ \mathrm d x = \frac {x^m \cosh a x} a - \frac m a \int x^{m - 1} \cosh a x \ \mathrm d x + C$

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 $x^m \cosh a x$