$\int_{-\frac{\pi}{8092}}^{\frac{\pi}{8092}} \frac{\sec (2023 x)}{1+(2023)^{(2023 x)}} d x=$

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_{\,\pi /6}^{\,\pi /3} {\,\frac{{dx}}{{1 + \sqrt {\cot x} }}} $ is equal to:

If $f(t) = \int_0^t \tan^{(2n-1)} x \, dx$,$n \in N$,then $f(t+\pi) =$

$\int_{5 \pi}^{25 \pi}|\sin 2 x+\cos 2 x| d x=$ (in $\sqrt{2}$)

If $I_1 = \int_0^{\pi / 4} \sin^2 x \, dx$ and $I_2 = \int_0^{\pi / 4} \cos^2 x \, dx$, then,