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

$\int_0^{\pi /2} {\frac{1}{{1 + \sqrt {\tan x} }}} \,dx = $

If $f(x) = f(2 - x)$,then $\int_{0.5}^{1.5} xf(x) dx$ equals

$\int_{0}^{1} \log \left(\frac{1}{x}-1\right) d x$ is equal to

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 $.....$

For $0 \le x \le \frac{\pi}{2}$,the value of $\int_{0}^{\sin^{2}x} \sin^{-1}(\sqrt{t}) \, dt + \int_{0}^{\cos^{2}x} \cos^{-1}(\sqrt{t}) \, dt$ is equal to