# Systematic method to evaluate $\int x(e^x) dx$

My calculus skills are pathetic. I haven't done the following integration exercise before, but this looks like it should easy to evaluate.

$$\int x e^x\ \ \mathrm{dx}$$

I can probably guess-and-check my way to the solution. But I am looking for a more systematic, algorithmic method to doing this. The first thing I tried was

$$\mathrm{\frac{d}{dx}}(x e^x) = e^x + xe^x \implies xe^x =\mathrm{\frac{d}{dx}}(x e^x)-e^x$$

but how to go from here to $\int x e^x\ \ \mathrm{dx}$, if that's even possible.

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I like your method. Try $xe^x$. The derivative is $xe^x+e^x$, wrong. But if we subtract $e^x$, that will fix it. So use $xe^x-e^x$. – André Nicolas Nov 14 '13 at 0:33
Ah I see. Should have been totally obvious. – NaN Nov 14 '13 at 0:36
Integration by parts is the standard method here. But there is a lot to be said for more informal methods. – André Nicolas Nov 14 '13 at 0:40