Question

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

Answer 1

\[Let I = \int_{- \frac{\pi}{2}}^\frac{\pi}{2} \sin^3 x\ d x\]

\[ = \int_{- \frac{\pi}{2}}^\frac{\pi}{2} \sin x \sin^2 x\ dx\]

\[ = \int_{- \frac{\pi}{2}}^\frac{\pi}{2} \sin x\left( 1 - \cos^2 x \right) dx\]

\[Let\ \cos x = t, then - \sin x\ dx = dt, \]

\[When\, x \to - \frac{\pi}{2} ; t \to 0\ and\ x \to \frac{\pi}{2} ; t \to 0\]

\[I = \int_0^0 \left( - 1 + t^2 \right) dt\]

\[\]

\[ = 0\]