Explanation

(a)
Let $I = \int_{0}^{\pi/2}{\frac{\sin^{2/3}x}{\sin^{2/3}{x + \cos^{2/3}x}}dx}$ …(i)
and $I = \int_{0}^{\pi/2}{\frac{\sin^{2/3}\left( \frac{\pi}{2} - x \right)}{{sin^{2/3}\left( \frac{\pi}{2} - x \right)}{+ \cos^{2/3}\left( \frac{\pi}{2} - x \right)}}dx}$
$\Rightarrow \ I = \int_{0}^{\pi/2}{\frac{\cos^{2/3}x}{\cos^{2/3}{x + \sin^{2/3}x}}dx}$ …(ii)
On adding Eqs.(i) and (ii), we get
$2I = \int_{0}^{\pi/2}\frac{(\sin^{2/3}{x + \cos^{2/3}x})}{(\sin^{2/3}{x + \cos^{2/3}x})}dx$
$\Rightarrow 2I = \int_{0}^{\pi/2}{1\ dx\ } \Rightarrow I = \frac{1}{2}\lbrack x\rbrack_{0}^{\pi/2}$
$\Rightarrow \ I = \frac{\pi}{4}$