Some Properties of Definite Integrals

The value of $\underset{0}{\overset{1}{\int }}{\tan }^{-1}\left(\frac{2x-1}{1+x-x^2}\right)dx$ is

If $f\left(a+b-x\right)=f\left(x\right),$ then $\underset{a}{\overset{b}{\int }}xf\left(x\right)dx$ is equal to

$\int \frac{\cos 2x}{{\left(\sin x+\cos x\right)}^2}dx$ is equal to

$\int \frac{dx}{e^x+e^{-x}}$ is equal to

Evaluate $\underset{0}{\overset{1}{\int }}{e}^{2-3x}dx$ as a limit of a sum.

Prove that $\underset{0}{\overset{1}{\int }}{\sin }^{-1}xdx=\frac{\pi }{2}-1$

Prove that $\underset{0}{\overset{\frac{\pi }{4}}{\int }}2 {\tan }^3 xdx=1-\log 2$

Prove that $\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\sin }^{3}xdx=\frac{2}{3}$

Prove that $\underset{-1}{\overset{1}{\int }}{x}^{17}{\cos }^{4}xdx=0$

Prove that $\underset{0}{\overset{1}{\int }}x{e}^{x}dx=1$

Prove that $\underset{1}{\overset{3}{\int }}\frac{dx}{x^2\left(x+1\right)}=\frac{2}{3}+\log \frac{2}{3}$

Evaluate the definite integral $\underset{1}{\overset{4}{\int }}\left[\left|x-1\right|+\left|x-2\right|+\left|x-3\right|\right]dx$

Evaluate the definite integral $\underset{0}{\overset{\pi }{\int }}\frac{x\tan x}{\sec x+\tan x}dx$

Evaluate the definite integral $\underset{0}{\overset{\frac{\pi }{2}}{\int }}\sin 2x{\tan }^{-1}\left(\sin x\right)dx$

Evaluate the definite integral $\underset{0}{\overset{\frac{\pi }{4}}{\int }}\frac{\sin x+\cos x}{9+16\sin 2x}dx$

Evaluate the definite integral $\underset{0}{\overset{1}{\int }}\frac{dx}{\sqrt{1+x}-\sqrt{x}}$

Evaluate the definite integral $\underset{\frac{\pi }{6}}{\overset{\frac{\pi }{3}}{\int }}\frac{\sin x+\cos x}{\sqrt{\sin 2x}}dx$

Evaluate the definite integral $\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{{\cos }^2 xdx}{{\cos }^2 x+4 {\sin }^2 x}$

Evaluate the definite integral $\underset{0}{\overset{\frac{\pi }{4}}{\int }}\frac{\sin x\cos x}{{\cos }^{4}x+{\sin }^{4}x}dx$

Evaluate the definite integral $\underset{\frac{\pi }{2}}{\overset{\pi }{\int }}{e}^x\left(\frac{1-\sin x}{1-\cos x}\right)dx$