Please help me with this intergral question I am really confused

Question

Answer 1

Hello Angie,

This problem has a shortcut to do it. So you don't actually have to take any antiderivatives!

Notice that if you substitute -x for x, arctan(-x) would be -arctan(x) while

1 + (-x)⁴ would be the same as 1 + (x)⁴. This means the integrand function has f(-x) = -f(x) which makes it an odd function (symmetric to the origin). What would the integral of an odd function from -pi to pi be?

Hint: Compare the integral from -pi to 0 and the integral from 0 to pi.