Question

How do you integrate `int x^2e^x dx` using integration by parts?

Answer 1

`intx^2e^xdx = x^2e^x - 2xe^x + 2e^x + c`

Explanation:

The integration by parts formula say

`intudv = uv - intvdu`

So, if we pick `u = x^2` and `dv = e^x` so we'll have `du = 2x` and `v = e^x`

`intx^2e^xdx = x^2e^x - int2xe^xdx`

Now we pick `u = x` and `dv = e^x`, so `v = e^x` and `du = 1`

`intx^2e^xdx = x^2e^x - 2(xe^x - inte^xdx)`

This last integral is tabled, so

`intx^2e^xdx = x^2e^x - 2xe^x + 2e^x + c`