Answer
$$\int x^{n} e^{a x} d x=\frac{x^{n} e^{a x}}{a}-\frac{n}{a} \int x^{n-1} e^{a x} d x$$
Work Step by Step
Given
$$\int x^{n} e^{a x} d x=\frac{x^{n} e^{a x}}{a}-\frac{n}{a} \int x^{n-1} e^{a x} d x$$
Use integration by parts , let
\begin{aligned}
u&= x^n \ \ \ \ \ \ &dv&= e^{a x} dx\\
du&=nx^{n-1} dx \ \ \ \ \ \ &v&=\frac{1}{a} e^{a x}
\end{aligned}
then
\begin{aligned}
\int x^{n} \ln x d x &= uv-\int vdu \\
&= \frac{1}{a} x^n e^{a x}-\frac{n}{a}\int x^{n-1} e^{a x}dx \\
\end{aligned}
