Question

How do you integrate `x^3e^x`?

Answer 1

`int \ x^3e^x \ dx = x^3e^x-3x^2e^x+6xe^x-6e^x+C`

Explanation:

Note that:

`d/(dx) (x^3e^x) = x^3e^x + 3x^2e^x`

`d/(dx) (-3x^2e^x) = -3x^2e^x-6xe^x`

`d/(dx) (6xe^x) = 6xe^x+6e^x`

`d/(dx) (-6e^x) = -6e^x`

So:

`d/(dx) (x^3e^x-3x^2e^x+6xe^x-6e^x) = x^3e^x`

So:

`int \ x^3e^x \ dx = x^3e^x-3x^2e^x+6xe^x-6e^x+C`