R D Sharma Solutions for Chapter: Definite Integrals, Exercise 8: REVISION EXERCISE

If integral of$\underset{0}{\overset{\pi /2}{\int }}\frac{x\sin x\cos x}{{\sin }^{4}x+{\cos }^{4}x}dx=\frac{{\pi }^2}{k}$, then the value of $k$ is

If integral of$\underset{0}{\overset{\pi /2}{\int }}\frac{{\cos }^{2}x}{\sin x+\cos x}dx=\frac{1}{\sqrt{k}}\log \left(\sqrt{2}+1\right)$, then the value of $k$ is

If integral of $\underset{0}{\overset{\pi }{\int }}\cos 2x\mathrm{logsin}xdx=-\frac{\pi }{k}$, then the required value of $k$ is 

If integral of $\underset{0}{\overset{\pi }{\int }}\frac{x}{a^2-{\cos }^{2}x}dx=\frac{{\pi }^2}{ka\sqrt{\left(a^2-1\right)}},a>1$, then the value of $k$ is

If integral of $\underset{0}{\overset{\pi }{\int }}\frac{x\tan x}{\sec x+\tan x}dx=\frac{\pi }{2}\left(\pi -k\right)$, then the value of $k$ is

Evaluate the integral $\underset{2}{\overset{3}{\int }}\frac{\sqrt{x}}{\sqrt{5-x}+\sqrt{x}}dx$

If integral of $\underset{0}{\overset{\pi /2}{\int }}\frac{{\sin }^{2}x}{\sin x+\cos x}dx=\frac{1}{\sqrt{k}}\log \left(\sqrt{2}+1\right)$, then the value of $k$ is

If integral of $\underset{0}{\overset{\pi /2}{\int }}\frac{x}{{\sin }^{2}x+{\cos }^{2}x}dx=\frac{{\pi }^2}{k}$, then the value of $k$ is