Question

Evaluate $\underset{-\pi /4}{\overset{\pi /4}{\int }}{x}^3{\sin }^{4}xdx$

Answer 1

$f\left(x\right)=x^3{\sin }^{4}x$
$\Rightarrow f(-x)=-x^3{\sin }^{4}x=-f\left(x\right)$
This implies that $f\left(x\right)$ is an odd function.
So, $\underset{-\pi /4}{\overset{\pi /4}{\int }}{x}^3{\sin }^{4}xdx=0$