Question

How do you find the antiderivative of `x[e^(x^2)]`?

Answer 1

Use substitution with `u = x^2`

Explanation:

I see that the question was posted under "Integration by Parts", but we can find this integral using u substitution.

`int x e^(x^2) dx`

Let `u = x^2`, o that `du = 2x dx` and `xdx = 1/2 du`

With this substitution, we get:

`int x e^(x^2) dx = 1/2 int e^u du`

`= 1/2 e^u +C`

`= 1/2e^(x^2) +C`

Check the answer by differentiating.

Note The integral `int xe^(5x)dx` does call for integration by parts, because the substitution `u = 5x` won't get us something we can integrate.

Note 2 I have assumed here that I can use "integral" in place of "antiderivative" without loss of understanding.