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Question
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Solution
\[Let I = \int_{- \frac{\pi}{2}}^\frac{\pi}{2} \sin^3 x\ d x\]
\[ = \int_{- \frac{\pi}{2}}^\frac{\pi}{2} \sin x \sin^2 x\ dx\]
\[ = \int_{- \frac{\pi}{2}}^\frac{\pi}{2} \sin x\left( 1 - \cos^2 x \right) dx\]
\[Let\ \cos x = t, then - \sin x\ dx = dt, \]
\[When\, x \to - \frac{\pi}{2} ; t \to 0\ and\ x \to \frac{\pi}{2} ; t \to 0\]
\[I = \int_0^0 \left( - 1 + t^2 \right) dt\]
\[\]
\[ = 0\]
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