By using properties of definite integrals, evaluate the integrals
$\int_{- \frac{π}{2}}^{\frac{π}{2}} s i n^{7} x d x .$

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Solution

$\int_{- \frac{π}{2}}^{\frac{π}{2}} s i n^{7} x d x .$
Here, $f (x) = s i n^{7} x f (- x) = s i n 7 (- x) = [- s i n x]^{7} = - s i n^{7} x ∴ f (- x) = - f (x) So, f(x) is an odd function . \int_{- \frac{π}{2}}^{\frac{π}{2}} s i n^{7} x d x . = 0 [∵ \int_{- a}^{a} f (x) d x = 0$ $if f(x) is an odd function$

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