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.

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As pointed out by [zz20s](https://math.stackexchange.com/users/213842/zz20s)
, the answer to this integral can be seen on the wikipedia page: [Sophomore's dream](https://en.wikipedia.org/wiki/Sophomore%27s_dream?wprov=sfla1) and the page accompanies a proof too