Computing $\int_{-1}^{1}\left(2|x|+x \arctan(2x^2) \cos(x^2) \right)dx $

Question

I am trying to compute $$\int_{-1}^{1}\left(2|x|+x \arctan(2x^2) \cos(x^2) \right)dx .$$

I started with $$ \int_{-1}^{1}2|x|dx + \int_{-1}^{1}x \arctan(2x^2) \cos(x^2) dx $$ $$ \int_{-1}^{0}-2x dx +\int_{0}^{1}2x dx+ \int_{-1}^{1}x \arctan(2x^2) \cos(x^2) dx $$ $$ 2+ \int_{-1}^{1}x \arctan(2x^2)\cos(x^2)dx $$ how to solve the last integral ?

Answer 1

Hint: If $f(x)$ is an odd function then $$\int_{-a}^{a} f(x) dx =0.$$