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

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Answer 1 · 2016-11-19T01:54:40

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

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

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

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