By using the properties of definite integrals,evaluate the integral $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^{7} x \, dx$.
(0) Let $I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^{7} x \, dx$ ..... $(1)$
Let $f(x) = \sin^{7} x$.
Check if the function is even or odd:
$f(-x) = \sin^{7}(-x) = (\sin(-x))^{7} = (-\sin x)^{7} = -\sin^{7} x = -f(x)$.
Since $f(-x) = -f(x)$,the function $f(x) = \sin^{7} x$ is an odd function.
According to the property of definite integrals,if $f(x)$ is an odd function,then $\int_{-a}^{a} f(x) \, dx = 0$.
Therefore,$I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^{7} x \, dx = 0$.
Let $f(x) = \sin^{7} x$.
Check if the function is even or odd:
$f(-x) = \sin^{7}(-x) = (\sin(-x))^{7} = (-\sin x)^{7} = -\sin^{7} x = -f(x)$.
Since $f(-x) = -f(x)$,the function $f(x) = \sin^{7} x$ is an odd function.
According to the property of definite integrals,if $f(x)$ is an odd function,then $\int_{-a}^{a} f(x) \, dx = 0$.
Therefore,$I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin^{7} x \, dx = 0$.
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