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Question
\[\int\limits_{- \pi/2}^{\pi/2} \sin^9 x dx\]
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Solution
\[\int_\frac{- \pi}{2}^\frac{\pi}{2} \sin^9 x d x\]
\[\text{Let }f(x) = \sin^9 x\]
\[\text{Consider, }f(-x) = \sin^9 \left( - x \right) = - \sin^9 x = - f\left( x \right)\]
Thus f(x) is an odd function
Therefore,
\[ \int_\frac{- \pi}{2}^\frac{\pi}{2} \sin^9 x d x = 0\]
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