Evaluate $\int_{-1}^{1} x^{3} \cos^{2} x\,dx$.

Explanation

1. 01

   $f(-x) = -f(x)$ for $f(x) = x^{3} \cos^{2} x$: the function is odd, since $\cos^2(-x) = \cos^2 x$ and $(-x)^3 = -x^3$.

2. 02

   The integral of an odd function over $[-a, a]$ is 0: the area below the axis cancels the area above it.