# 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:

 $\displaystyle u$ $=$ $\displaystyle x$ $\displaystyle \implies \ \$ $\displaystyle \frac {\mathrm d u}{\mathrm d x}$ $=$ $\displaystyle 1$ Derivative of Identity Function

and let:

 $\displaystyle \frac {\mathrm d v}{\mathrm d x}$ $=$ $\displaystyle \sinh a x$ $\displaystyle \implies \ \$ $\displaystyle v$ $=$ $\displaystyle \frac {\cosh a x} a$ Primitive of $\sinh a x$

Then:

 $\displaystyle \int x \sinh a x \ \mathrm d x$ $=$ $\displaystyle x \left({\frac {\cosh a x} a}\right) - \int \left({\frac {\cosh a x} a}\right) \times 1 \ \mathrm d x + C$ Integration by Parts $\displaystyle$ $=$ $\displaystyle \frac {x \cosh a x} a - \frac 1 a \int \cosh a x \ \mathrm d x + C$ Linear Combination of Integrals $\displaystyle$ $=$ $\displaystyle \frac {x \cosh a x} a - \frac 1 a \left({\frac {\sinh a x} a}\right) + C$ Primitive of $\cosh a x$ $\displaystyle$ $=$ $\displaystyle \frac {x \cosh a x} a - \frac {\sinh a x} {a^2} + C$ simplification

$\blacksquare$