Primitive of Hyperbolic Sine of a x over Power of x

Theorem

 * $\displaystyle \int \frac {\sinh a x \ \mathrm d x} {x^n} = \frac {-\sinh a x} {\left({n - 1}\right) x^{n - 1} } + \frac a {n - 1} \int \frac {\cosh a x \ \mathrm d x} {x^{n - 1} } + C$

Proof
With a view to expressing the problem 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 $\dfrac {\cosh a x} {x^n}$