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

Question

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.

Answer 1

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

Answer 2

Have you considered Integration by Parts?

Answer 3

Use Integration by Parts. Essentially -

Let $u$ and $v$ be two functions of x. Then Integration By Parts says that -

$$\int uvdx = u\int vdx - \int u' \bigg(\int vdx \bigg)dx$$

• $u'$ being the derivative of $u$.

Now you need to be very strategic in choosing which function should be $v$ and $u$ because its always $v$ that is being integrated. I'll leave that as an exercise for you in the example you gave.