Find $\int{x^2chxdx}$ where $chx$ is hyperbolic cosine.

Question

Find $\int{x^2chxdx}$ where $chx$ is hyperbolic cosine.

My steps.

$\int x^2chxdx$ = $x^2shx-\int{shxdx^2} = x^2shx-\int{shx2xdx}= x^2shx-2\int{xshxdx}$

I can't continue from here.How find $\int xshxdx$

Answer 1

Integrate by parts again. We have

$$\int x^2\cosh(x)dx=x^2\sinh(x)-2\int x\sinh(x)dx$$ $$=x^2\sinh(x)-2\left[x\cosh(x)-\int\cosh(x)dx\right]$$ $$=x^2\sinh(x)-2x\cosh(x)+2\sinh(x)+C$$