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

Apr 30, 2016

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

$\int {x}^{2} {e}^{{x}^{2}} \mathrm{dx}$

In order to integrate ${e}^{{x}^{2}} \mathrm{dx}$ we need an $x$ so that we can use substitution.

$\int {x}^{2} {e}^{{x}^{2}} \mathrm{dx} = \int x {e}^{{x}^{2}} x \mathrm{dx}$ .

Let $u = x$ and $\mathrm{dv} = {e}^{{x}^{2}} x \mathrm{dx}$

The $\mathrm{du} = 1 \mathrm{dx}$ and $v = \frac{1}{2} {e}^{{x}^{2}}$

$\int {x}^{2} {e}^{{x}^{2}} \mathrm{dx} = \frac{1}{2} x {e}^{{x}^{2}} - \frac{1}{2} \int {e}^{{x}^{2}} \mathrm{dx}$.

Now we need to stop.

$\int {e}^{{x}^{2}} \mathrm{dx}$ 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