Let $I = \int\limits_{0}^{\pi/2}\frac{\cos^5 x}{\sin^5 x + \cos^5 x}dx$ ...(i) Also, $I = \int\limits_{0}^{\pi/2}\frac{\cos^5(\pi/2 - x)}{\sin^5(\pi/2 - x) + \cos^5(\pi/2 - x)}dx$ $\left[\because \int\limits_{0}^{a} f(x) dx = \int\limits_{0}^{a} f(a-x) dx\right]$ $= \int\limits_{0}^{\pi/2}\frac{\sin^5 x}{\cos^5 x + \sin^5 x}dx$ ...(ii) Adding (i) and (ii), we have $2I = \int\limits_{0}^{\pi/2}\frac{\cos^5 x + \sin^5 x}{\cos^5 x + \sin^5 x}dx$ $= \int\limits_{0}^{\pi/2} dx$ $= [x]_{0}^{\pi/2} = \pi/2$ Hence, $I = \pi/4$