TITLE: What is $\int x^re^xdx$?
QUESTION [6 upvotes]: Is there any simple way to get integral of $e^{x}x^{r}, r \in \mathbb{R}$? 
Basically I want to solve this: 
$$\displaystyle \int \frac{e^t(4t^2+1)}{2t \sqrt{t}}dt$$
so I will appreciate any help in both of the above problems.

REPLY [8 votes]: Hint: Break up the integral into two integrals
$$
\int \frac{e^t(4t^2 + 1)}{2t \sqrt{t}} dt = \int 2e^t \sqrt{t} dt + \int \frac{e^t}{2t^{3/2}}dt
$$
Then use integration by parts on one of them until you have a constant multiple of the other, and see what happens.
Note: In general, we can't find $\int e^x x^r dx$. The definite integral $\int_0^\infty x^r e^x dx$ is (almost) the gamma function.