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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
2  
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

2 Answers 2

Such a systematic method exists and it's called integration by parts.

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You beat me by < 1 minute! –  Ahaan S. Rungta Nov 14 '13 at 0:29

Have you considered Integration by Parts?

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