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

Jan 24, 2017

$\int {x}^{2} {e}^{2 x} \mathrm{dx} = {e}^{2 x} / 4 \left(2 {x}^{2} - 2 x + 1\right) + C$

#### Explanation:

Take ${x}^{2}$ as finite part, so in the integration by parts the degree of $x$ decreases:

$\int {x}^{2} {e}^{2 x} \mathrm{dx} = \frac{1}{2} \int {x}^{2} d \left({e}^{2 x}\right) = \frac{{x}^{2} {e}^{2 x}}{2} - \int x {e}^{2 x} \mathrm{dx}$

We can now solve the resulting integral by parts again:

$\int x {e}^{2 x} \mathrm{dx} = \frac{1}{2} \int x d \left({e}^{2 x}\right) = \frac{x {e}^{2 x}}{2} - \frac{1}{2} \int {e}^{2 x} \mathrm{dx}$

and we can now solve the last integral directly:

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

Putting it all together:

$\int {x}^{2} {e}^{2 x} \mathrm{dx} = \frac{{x}^{2} {e}^{2 x}}{2} - \frac{x {e}^{2 x}}{2} + \frac{1}{4} {e}^{2 x} + C$