What is the antiderivative of $xe^x$ ?

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Answer 1 · 2016-07-08T07:39:11

Let $u=x$ and $dv= e^(x) dx$ , so $du=dx$ and $v=e^(x)$

Knowing that

$int u dv = uv - int v du$

You can see that

$int x e^(x) dx$
$= xe^(x)-int e^x dx$
$= xe^(x)-e^(x)+C$

$=e^(x)(x-1)+C$

In this case, you'd have to perform integration by parts, which is given by the following formula:

$int u dv = uv - int v du$

Note that if you would have chosen $u=e^(x)$ and $dv = x dx$ , your integral would be even more complicated.

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