Each entry of List-I is a definite integral or a limit of sums, and each has an exact value. Find each one, and match it with its value in List-II. Some entries of List-II may be used more than once, and some not at all. Each entry of List-I needs a different idea: a limit of sums, a symmetry of the interval, integration by parts, and a square root that hides a modulus. List-I P. $\displaystyle\lim_{n \to \infty} \sum_{k=1}^{n} \frac{1}{\sqrt{n^2 + nk}}$ Q. $\displaystyle\int_0^{\pi/2} \sin 2x\,\ln(\tan x)\,dx$ R. $\displaystyle\int_0^{1} \tan^{-1} x\,dx$ S. $\displaystyle\int_0^{\pi/2} \sqrt{1 - \sin 2x}\,dx$ List-II 1. $0$ 2. $2(\sqrt2 - 1)$ 3. $\sqrt2 - 1$ 4. $\dfrac{\pi}{4} - \dfrac12\ln 2$ 5. $\dfrac{\pi}{4} + \dfrac12\ln 2$