$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sqrt{\cos x}}{\sqrt{\sin x}+\sqrt{\cos x}} d x$ is equal to . . . . . . .

The value of $\int_0^1 \tan^{-1}(1-x+x^2) dx$ is

If $\int_0^{\frac{\pi}{2}} \log \cos x \, dx = \frac{\pi}{2} \log \left(\frac{1}{2}\right)$,then $\int_0^{\frac{\pi}{2}} \log \sec x \, dx = $

Let $f: R \rightarrow R$ be a function defined by $f(x)=\frac{4^x}{4^x+2}$ and $M=\int_{f(a)}^{f(1-a)} x \sin^4(x(1-x)) dx,$ $N=\int_{f(a)}^{f(1-a)} \sin^4(x(1-x)) dx;$ $a \neq \frac{1}{2}.$ If $\alpha M=\beta N,$ $\alpha, \beta \in N,$ then the least value of $\alpha^2+\beta^2$ is equal to $.....$

$\int_{0}^{\pi} \frac{\cos ^{4} x}{\cos ^{4} x+\sin ^{4} x} d x$ is equal to

The value of the integral $\int_{-\pi / 2}^{\pi / 2} \frac{\sin^2 x}{1+e^x} \, dx$ is