$ \int_{0}^{\pi / 4} \log \left(\frac{\sin x+\cos x}{\cos x}\right) d x $

If $[a]$ denotes the greatest integer which is less than or equal to $a$, then the value of the integral $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}[\sin x \cos x] dx$ is

The value of $\int\limits_{ - \pi /2}^{\pi /2} {\frac{{{{\sin }^2}x}}{{1 + {2^x}}}dx} $ is

The value of $\int_0^{\pi / 2} \frac{(\cos x)^{\sin x}}{(\cos x)^{\sin x}+(\sin x)^{\cos x}} d x$ is

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 $.....$

The integral $\int\limits_0^{\frac{1}{2}} \frac{\ln(1 + 2x)}{1 + 4x^2} dx$ equals