Answer
$$x\cosh x - \sinh x + C$$
Work Step by Step
$$\eqalign{
& \int {x\sinh x} dx \cr
& {\text{substitute }}u = x,{\text{ }}du = dx \cr
& dv = \sinh xdx,{\text{ }}v = \cosh x \cr
& {\text{applying integration by parts}} \cr
& \int {udv} = uv - \int {vdu} \cr
& {\text{, we have}} \cr
& = \left( x \right)\left( {\cosh x} \right) - \int {\left( {\cosh x} \right)dx} \cr
& {\text{multiply}} \cr
& = x\cosh x - \int {\cosh x} dx \cr
& {\text{find the antiderivative}} \cr
& = x\cosh x - \sinh x + C \cr} $$
