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