Question:
$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin ^{7} x d x$
Solution:
Let $I=\int_{\frac{\pi}{2}}^{\pi} \sin ^{7} x d x$ ...(1)
As $\sin ^{7}(-x)=(\sin (-x))^{7}=(-\sin x)^{7}=-\sin ^{7} x$, therefore, $\sin ^{2} x$ is an odd function.
It is known that, if $f(x)$ is an odd function, then $\int_{-a}^{a} f(x) d x=0$
$\therefore I=\int_{\frac{\pi}{2}}^{\frac{\pi}{2}} \sin ^{7} x d x=0$
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