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

## Theorem

$\ds \int x \sinh^2 a x \rd x = \dfrac {x \sinh 2 a x} {4 a} - \frac {\cosh 2 a x} {8 a^2} - \frac {x^2} 4 + C$

## 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:

 $\ds u$ $=$ $\ds x$ $\ds \leadsto \ \$ $\ds \frac {\d u} {\d x}$ $=$ $\ds 1$ Derivative of Identity Function

and let:

 $\ds \frac {\d v} {\d x}$ $=$ $\ds \sinh^2 a x$ $\ds \leadsto \ \$ $\ds v$ $=$ $\ds \frac {\sinh 2 a x} {4 a} - \frac x 2$ Primitive of $\sinh^2 a x$

Then:

 $\ds \int x \sinh^2 a x \rd x$ $=$ $\ds x \paren {\frac {\sinh 2 a x} {4 a} - \frac x 2} - \int \paren {\frac {\sinh 2 a x} {4 a} - \frac x 2} \times 1 \rd x + C$ Integration by Parts $\ds$ $=$ $\ds \frac {x \sinh 2 a x} {4 a} - \frac {x^2} 2 - \frac 1 {4 a} \int \sinh 2 a x \rd x + \frac 1 2 \int x \rd x + C$ Linear Combination of Integrals $\ds$ $=$ $\ds \frac {x \sinh 2 a x} {4 a} - \frac {x^2} 2 - \frac 1 {4 a} \int \sinh 2 a x \rd x + \frac {x^2} 4 + C$ Primitive of Power $\ds$ $=$ $\ds \frac {x \sinh 2 a x} {4 a} - \frac {x^2} 2 - \frac 1 {4 a} \frac {\cosh 2 a x} {2 a} + \frac {x^2} 4 + C$ Primitive of $\sinh a x$ $\ds$ $=$ $\ds \dfrac {x \sinh 2 a x} {4 a} - \frac {\cosh 2 a x} {8 a^2} - \frac {x^2} 4 + C$ simplifying

$\blacksquare$