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

Aug 22, 2015

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.

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

Let $u = {x}^{2}$, o that $\mathrm{du} = 2 x \mathrm{dx}$ and $x \mathrm{dx} = \frac{1}{2} \mathrm{du}$

With this substitution, we get:

$\int x {e}^{{x}^{2}} \mathrm{dx} = \frac{1}{2} \int {e}^{u} \mathrm{du}$

$= \frac{1}{2} {e}^{u} + C$

$= \frac{1}{2} {e}^{{x}^{2}} + C$

Note The integral $\int x {e}^{5 x} \mathrm{dx}$ does call for integration by parts, because the substitution $u = 5 x$ won't get us something we can integrate.