Answer
$${x^2}\sinh x - 2x\cosh x + 2\sinh x + C$$
Work Step by Step
$$\eqalign{
& \int {{x^2}\cosh x} dx \cr
& u = {x^2},{\text{ }}du = 2xdx \cr
& dx = \cosh xdx,{\text{ }}v = \sinh x \cr
& {\text{integration by parts }} = uv - \int {vdu} \cr
& = {x^2}\sinh x - \int {\sinh x\left( {2x} \right)dx} \cr
& = {x^2}\sinh x - 2\int {x\sinh xdx} \cr
& u = x,{\text{ }}du = dx \cr
& dx = \sinh xdx,{\text{ }}v = \cosh xdx \cr
& = {x^2}\sinh x - 2\left( {x\cosh x - \int {\left( {\cosh x} \right)\left( {dx} \right)} } \right) \cr
& = {x^2}\sinh x - 2\left( {x\cosh x - \int {\cosh xdx} } \right) \cr
& = {x^2}\sinh x - 2x\cosh x + 2\int {\cosh xdx} \cr
& = {x^2}\sinh x - 2x\cosh x + 2\sinh x + C \cr} $$
