Suppose $\displaystyle P$ is a polynomial of degree $\displaystyle k\geq 1$. Prove that

$\displaystyle \int P(x)e^x dx = e^x\sum_{j=0}^{k}(-1)^jP^{(j)}(x)+C$

by induction on $\displaystyle k$. Here, $\displaystyle P^{(j)}$ is the $\displaystyle jth$ derivative of $\displaystyle P$, and $\displaystyle P^{(0)}$ is to be interpreted as $\displaystyle P$.

A hint I was provided was to use integration by parts.