Question

How do you use integration by parts to find `intxe^x dx`?

Answer 1

To integrate by parts, we have to pick a `u` and `dv` such that

`int u dv= uv-int v du`

We can pick what `u` and `dv` are, though we should typically try to pick `u` such that `du` is "simpler". (By the way, `du` just means the derivative of `u`, and `dv` just means derivative of `v`)

So, from `int u dv`, let's pick `u=x` and `dv=e^x`. It is helpful to fill in all of the parts you will use throughout the problem before you start integrating:

`u=x`
`(du)/dx=1` so `du=1dx=dx`

`dv=e^x`
`v=int dv= int e^x dx =e^x`

Now let's plug everything back into the original formula.

`int u dv= uv-int v du`
`int xe^x dx= xe^x-int e^xdx`
You can integrate the last part quite simply now, to get:

`int xe^x dx= xe^x-int e^xdx= xe^x-e^x +c`