Question

Evaluate : ${\int }_0^{\pi /2}{\cos }^{3}xdx$

Answer 1

$I=\stackrel{\frac{\pi }{2}}{\underset{0}{\int }}{\cos }^{3}xdx$

$I=\stackrel{\frac{\pi }{2}}{\underset{0}{\int }}{\cos }^{2}x.\cos xdx=\stackrel{\frac{\pi }{2}}{\underset{0}{\int }}(1-{\sin }^{2}x).\cos xdx$

Let $u=\sin x;dx=\cos xdx$

$I=\stackrel{\frac{\pi }{2}}{\underset{0}{\int }}(1-u^2)dx={[u-\frac{u^3}{3}]}_0^{\frac{\pi }{2}}$

$I={[\sin x-\frac{{\sin }^{3}x}{3}]}_0^{\frac{\pi }{2}}=1-\frac{1}{3}$

$I=\frac{2}{3}$.