By using the properties of definite integrals, evaluate the integral $\int_{- \frac{π}{2}}^{\frac{π}{2}} {sin}^{7} x d x$

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Solution

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

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