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

Nov 19, 2016

$\int {x}^{n} {e}^{{x}^{n}} \mathrm{dx} = \frac{1}{n} x {e}^{{x}^{n}} + \frac{x \Gamma \left(\frac{1}{n} , - {x}^{n}\right)}{n {\left(- {x}^{n}\right)}^{\frac{1}{n}}} + C$

#### Explanation:

We know that

$\frac{d}{\mathrm{dx}} \left(x {e}^{{x}^{n}}\right) = n {x}^{n} {e}^{{x}^{n}} + {e}^{{x}^{n}}$ so

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

The integral $\int {e}^{{x}^{n}} \mathrm{dx}$ is a manual integral and is equal to

$\int {e}^{{x}^{n}} \mathrm{dx} = - \frac{x \Gamma \left(\frac{1}{n} , - {x}^{n}\right)}{n {\left(- {x}^{n}\right)}^{\frac{1}{n}}} + C$ so

$\int {x}^{n} {e}^{{x}^{n}} \mathrm{dx} = \frac{1}{n} x {e}^{{x}^{n}} + \frac{x \Gamma \left(\frac{1}{n} , - {x}^{n}\right)}{n {\left(- {x}^{n}\right)}^{\frac{1}{n}}} + C$