$\int_{0}^{\frac{\pi}{2}} \frac{1-\cot x}{\operatorname{cosec} x+\cos x} d x=$

If $\alpha=\int_{\frac{1}{2}}^2 \frac{\tan ^{-1} x}{2 x^2-3 x+2} d x$,then the value of $\sqrt{7} \tan \left(\frac{2 \alpha \sqrt{7}}{\pi}\right)$ is $....$. (Here,the inverse trigonometric function $\tan ^{-1} x$ assumes values in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.)

Value of the definite integral,$\int\limits_{ - \frac{1}{2}}^{\frac{1}{2}} {\,(\,\,{{\sin }^{ - 1}}(3x - 4{x^3})\,\, - \,\,{{\cos }^{ - 1}}(4{x^3} - 3x)\,\,)\,dx} \,\,$

Evaluate $\int_{0}^{\pi} \frac{x \, dx}{a^{2} \cos ^{2} x+b^{2} \sin ^{2} x}$

If $\int_{-a}^{a} (|x| + |x-2|) dx = 22$,$(a > 2)$ and $[x]$ denotes the greatest integer $\leq x$,then $\int_{a}^{-a} (x + [x]) dx$ is equal to ...........

The value of the definite integral $\int\limits_0^{\frac{\pi }{2}} {\sqrt {\tan x} \,dx} $ is