Question

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

Answer 1

See the explanation section below.

Explanation:

To integrate `x` to a power times `e` to a power, we expect to differentiate the `x` and integrate the `e` to a power

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

In order to integrate `e^(x^2) dx` we need an `x` so that we can use substitution.

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

Let `u = x` and `dv = e^(x^2)x dx`

The `du = 1 dx` and `v = 1/2 e^(x^2)`

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

Now we need to stop.

`int e^(x^2) dx` has no closed form solution using elementary functions. The integral has a name and some series approximations, but that's the best we can do.

You can read more about it here at Wolfram and here at Wikipedia