Explanation

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