Integrating $\int^2_{-2}\frac{x^2}{1+5^x}$
Solution 1:
$$\tag1I=\int_{-2}^{2}\frac{x^2}{1+5^x}dx$$ Note that $$\int_a^bf(x)dx=\int_a^bf(a+b-x)dx$$ Thus, $$\tag2I=\int_{-2}^{2}\frac{(-2+2-x)^2}{1+5^{-2+2-x}}dx=\int_{-2}^{2}\frac{x^2}{1+5^{-x}}dx=\int_{-2}^{2}\frac{5^xx^2}{1+5^{x}}dx$$
Add $(1)$ and $(2)$.
Solution 2:
Hint:
$$\frac1{1+5^{-x}} + \frac1{1+5^x} = 1$$