Question

\[\int\limits_{- \pi/2}^{\pi/2} \sin^3 x\ dx\]

Answer 1

\[Let I = \int_\frac{- \pi}{2}^\frac{\pi}{2} \sin^3 x d x\]
\[ = \int_\frac{- \pi}{2}^\frac{\pi}{2} \sin x\left( 1 - \cos^2 x \right)dx\]
\[ = \int_\frac{- \pi}{2}^\frac{\pi}{2} \sin x dx - \int_\frac{- \pi}{2}^\frac{\pi}{2} \sin x \cos^2 x dx\]
\[ = \left[ - \cos x \right]_\frac{- \pi}{2}^\frac{\pi}{2} + \left[ \frac{\cos^3 x}{3} \right]_{- \frac{\pi}{2}}^\frac{\pi}{2} \]
\[ = 0 + 0 = 0\]