# Primitive of Power of x by Hyperbolic Sine of a x

## Theorem

$\ds \int x^m \sinh a x \rd x = \frac {x^m \cosh a x} a - \frac m a \int x^{m - 1} \cosh a x \rd x + C$

## 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:

 $\ds u$ $=$ $\ds x^m$ $\ds \leadsto \ \$ $\ds \frac {\d u} {\d x}$ $=$ $\ds m x^{m - 1}$ Derivative of Power

and let:

 $\ds \frac {\d v} {\d x}$ $=$ $\ds \sinh a x$ $\ds \leadsto \ \$ $\ds v$ $=$ $\ds \frac {\cosh a x} a$ Primitive of $\sinh a x$

Then:

 $\ds \int x^m \sinh a x \rd x$ $=$ $\ds x^m \paren {\frac {\cosh a x} a} - \int \paren {\frac {\cosh a x} a} \paren {m x^{m - 1} } \rd x + C$ Integration by Parts $\ds$ $=$ $\ds \frac {x^m \cosh a x} a - \frac m a \int x^{m - 1} \cosh a x \rd x + C$ Primitive of Constant Multiple of Function

$\blacksquare$