Answer
$$\int {{x^n}\sin axdx} = - \frac{{{x^n}\cos ax}}{a} + \frac{n}{a}\int {{x^{n - 1}}\cos axdx} $$
Work Step by Step
$$\eqalign{
& \int {{x^n}\sin axdx} \cr
& {\text{substitute }}u = {x^n},{\text{ }}du = n{x^{n - 1}}dx \cr
& dv = \sin axdx,{\text{ }}v = - \frac{1}{a}\cos ax \cr
& {\text{applying integration by parts}} \cr
& \int {udv} = uv - \int {vdu} \cr
& {\text{, we have}} \cr
& \int {{x^n}\sin axdx} = \left( {{x^n}} \right)\left( { - \frac{1}{a}\cos ax} \right) - \int {\left( { - \frac{1}{a}\cos ax} \right)\left( {n{x^{n - 1}}dx} \right)} \cr
& {\text{simplify}} \cr
& \int {{x^n}\sin axdx} = - \frac{{{x^n}}}{a}\cos ax - \int {\left( { - \frac{{n{x^{n - 1}}}}{a}\cos ax} \right)dx} \cr
& \int {{x^n}\sin axdx} = - \frac{{{x^n}\cos ax}}{a} + \frac{n}{a}\int {{x^{n - 1}}\cos axdx} \cr} $$
