Explanation

(a)
Let $I = \int_{\pi/6}^{\pi/3}{\frac{1}{1 + \tan^{3}x}dx}$
$= \int_{\pi/6}^{\pi/3}{\frac{\cos^{3}x}{\sin^{3}{x + \cos^{3}x}}dx}$ ...(i)
$\Rightarrow I = \int_{\pi/6}^{\pi/3}{\frac{\sin^{3}x}{\sin^{3}{x + \cos^{3}x}}dx}\$ …(ii)
On adding Eqs. (i) and (ii), we get
$2I = \int_{\pi/6}^{\pi/3}{1dx = \lbrack x\rbrack_{\pi/6}^{\pi/3} = \frac{\pi}{6}}$
$\Rightarrow \ I = \frac{\pi}{12}$