Question

How do you evaluate the integral `int x^3e^(-x^2) dx` from 0 to `oo` if it converges?

Answer 1

You'll need to integrate by parts to evaluate.

Explanation:

`int x^3e^(-x^2) dx = -1/2int underbrace(x^2)_u underbrace(e^(-x^2) (-2)x dx)_(dv)`

`= -1/2 x^2 e^(-x^2)+ int x e^(x^2) dx`

`= -1/2 x^2 e^(-x^2)+ (-1/2) int (-2) x e^(x^2) dx`

`= -1/2x^2 e^(-x^2) - 1/2 e^(-x^2)`

(Of course `+C`, but we're going to use it for a definite integral, so I'll leave that off.)

Evaluate from `0` to `b`, then take the limit as `brarroo` to get

`int_0^oo x^3e^(-x^2) dx = 1/2`