Question

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

Answer 1

`int x^n e^(x^n)dx= 1/nx e^(x^n)+(x Gamma(1/n, -x^n))/(n (-x^n)^(1/n) )+C`

Explanation:

We know that

`d/dx(xe^(x^n))=nx^n e^(x^n)+e^(x^n)` so

`int x^n e^(x^n)dx = 1/nx e^(x^n)-int e^(x^n)dx`

The integral `int e^(x^n)dx` is a manual integral and is equal to

`int e^(x^n)dx=-(x Gamma(1/n, -x^n))/(n (-x^n)^(1/n) )+C` so

`int x^n e^(x^n)dx= 1/nx e^(x^n)+(x Gamma(1/n, -x^n))/(n (-x^n)^(1/n) )+C`