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

On the real line $R$,we define two functions $f$ and $g$ as follows:
$f(x) = \min \{x - [x], 1 - x + [x]\}$
$g(x) = \max \{x - [x], 1 - x + [x]\}$
where $[x]$ denotes the greatest integer function. The positive integer $n$ for which $\int_0^n (g(x) - f(x)) \, dx = 100$ is:

If $I = \int_0^{\frac{\pi}{4}} \log (1 + \tan x) \, dx$,then the value of $I$ is

If $f(x) = \frac{2 - x \cos x}{2 + x \cos x}$ and $g(x) = \ln x$ for $x > 0$,then the value of the integral $\int_{-\pi/4}^{\pi/4} g(f(x)) dx$ is

$\int_0^1 f(1 - x) \, dx$ has the same value as which of the following integrals?

$\int_0^\pi \frac{x \sin x}{1+\cos ^2 x} d x=$