# Thread: Hard integral from easy function

1. ## Hard integral from easy function

Hi!

I want to calculate the integral from x^x dx or x^(1/x) dx, but I don't know how I can start it. I would appreciate for any help or maybe if someone have a book with solved integrals like this two, then he could give a title or link to this.

2. Originally Posted by fliker
Hi!

I want to calculate the integral from x^x dx or x^(1/x) dx, but I don't know how I can start it. I would appreciate for any help or maybe if someone have a book with solved integrals like this two, then he could give a title or link to this.
You can do $\displaystyle \int_0^1 x^x\text{ }dx$ but there is no anti-derivative.

3. How calculate definite?

4. Originally Posted by fliker
How calculate definite?
$\displaystyle \int_0^1 x^x\text{ }dx=\int_0^1 e^{x\ln(x)}\text{ }dx=\int_0^1\sum_{n=1}^{\infty}\frac{x^n\ln^n(x)}{ n!}$$\displaystyle =\sum_{n=1}^{\infty}\int_0^1 \frac{x^n\ln^n(x)}{n!}=...$ etc.