$\int\limits_{ - \pi /2}^{\pi /2} {{{\sin }^7}x} dx$

Let $f\left( x \right) = {\sin ^7}x$

$\Rightarrow$ $f\left( { - x} \right) = {\left[ {\sin \left( { - x} \right)} \right]^7} = {\left( { - \sin \left( x \right)} \right)^7} = - f\left( x \right)$

$\Rightarrow$ $f\left( x \right)$ is an odd function of x.

$\Rightarrow$ $\int\limits_{ - \cfrac{\pi }{2}}^{\cfrac{\pi }{2}} {{{\sin }^7}xdx = 0}$