Let $f(x) = \sin^7x.$$\sin x$ is an odd functioni.e., if $h(x) = \sin x$ $\Rightarrow h(-x) = \sin(-x)$ $= -\sin(x) = -h(x)$⇒ odd power of sin $x$ is odd ⇒ $f(x)$ is an odd function of $x$.$\Rightarrow\int\limits_{-\pi/2}^{\pi/2}\sin^7xdx=0$ [ ∵ If $f(x)$ is odd $\Rightarrow\int\limits_{-a}^{a}f(x)dx=0$]