Question

What is the antiderivative of `xe^x`?

Answer 1

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`

Explanation:

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.