By using the properties of definite integrals, evaluate the integ… - Vidyadip

Question

By using the properties of definite integrals, evaluate the integral $\int\limits_{\frac{{ - \pi }}{2}}^{\frac{\pi }{2}} {{{\sin }^7}xdx} $

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Answer

Let $I = \int\limits_{\frac{{ - \pi }}{2}}^{\frac{\pi }{2}} {{{\sin }^7}xdx} $
Here $f(x) = \sin^7x$
$\therefore f\left( { - x} \right) = {\sin ^7}\left( { - x} \right)$
$(-\sin x)^7$
$= -\sin^7x = -f(x)$
$\therefore f(x)$ is an odd function of $x.$
$\therefore I = \int\limits_{\frac{{ - \pi }}{2}}^{\frac{\pi }{2}} {{{\sin }^7}xdx} = 0$
${\left[ {\because \int\limits_{ - a}^a {f\left( x \right)dx = 0} } \right.}$ when $f(x)$ is an odd function$].$

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