<s>[INST] Find $\int^{1}_{0}x^{x}dx$ The answer can be in the form of two defined constants:
$$A = \frac{1}{1^1}+\frac{1}{2^2} + \frac{1}{3^3} + \cdots$$
$$B = \frac{1}{2^2} + \frac{1}{4^4} + \frac{1}{6^6} + \cdots $$
I would highly appreciate it if you included your thought process in the answer itself.

As pointed out by zz20s
, the answer to this integral can be seen on the wikipedia page: Sophomore's dream and the page accompanies a proof too
 [/INST] I have written a detailed explanation to find the integral of a general expression of the form $f(x) =x^{cx^a} $ from $0$ to $1$ here. 

$$\int_{0}^{1} x^{cx^a}\, dx = 1- \frac{c}{(a+1)^2} + \frac{c^2}{(2a+1)^3} - \frac{c}{(3a+1)^4}+\ldots$$

On substituting $c =1$ and $a=1$, we get the desired result. 
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