$\int_0^{\pi / 2} \log _e(\sin 2 x) d x$

$\int_{\pi/6}^{\pi/3} \frac{dx}{1+\sqrt{\cot x}} = $ . . . . . . .

$\int_{-1/2}^{1/2} (\cos x) \left[ \log \left( \frac{1-x}{1+x} \right) \right] dx = $

$\int_{0}^{\pi} x f(\sin x) dx = $

ધારો કે $I=\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{1}{2-\cos 2 x}\left(\frac{3}{\pi}+\log \left(\frac{4+\sin x}{4-\sin x}\right)\right) d x$. આપેલ છે કે $\int \frac{d x}{1+k x^2}=\frac{1}{\sqrt{k}} \tan ^{-1}(\sqrt{k} x)+c, \tan ^{-1}(0)=0$ અને $\tan ^{-1}(\sqrt{3})=\frac{\pi}{3}$. તો $3 I^2=$

$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left(\frac{1+\sin^{2} x}{1+\pi^{\sin x}}\right) \, dx$ નું મૂલ્ય શોધો.