I am attempting to solve this integral/problem I found on brilliant. Given$$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\dfrac{x^{2}\cos(x)}{1+\exp(x^{2}\sin(x))}\,dx $$ converges to $\dfrac{\pi^{a}-b}{c}$, with $a,b,c \in \mathbb{Z}$, find $a+b+c.$ I was overly curious and spoiled the surprise by going to wolframalpha, but I would like some hints to solve the problem analytically without this helpful computation engine.
https://www.wolframalpha.com/input/?i=0.467401&assumption=%22ClashPrefs%22+-%3E+%7B%22Math%22%7D
Link $1$ is the evaluated integral, Link $2$ gives the answer in terms of $\pi$ and we let $a=2, b=8, c=4$ and the sum is $14.$ All hints to get initially started towards a solution are greatly appreciated. I tried a $u$ substitution and it was absolutely bologna. Feynman integration or by parts perhaps? Please help!
Best Answer
Note that this is of the form $$\int_{-a}^a\frac{\text{even(x)}}{1+b^{\text{odd}(x)}}\mathrm{d}x=\int_0^a \text{even}(x)\mathrm{d}x$$ for arbitrary even/odd functions and constants $a,b\in\mathbb{R}^+$. A proof of the above can be found here.