$\int_0^{\pi /2} \frac{\sin^{3/2} x}{\cos^{3/2} x + \sin^{3/2} x} dx = $

If $F(x) = f(x) + f\left(\frac{1}{x}\right)$,where $f(x) = \int_{1}^{x} \frac{\log_{e} t}{1+t} dt$,then $F(e) = $

If the value of $\int_{0}^{\pi/2} \sin^{4}(x) \cdot \cos^{2}(x) dx = \frac{\pi}{32}$,then the value of $\int_{0}^{\pi/2} \cos^{4}(x) \cdot \sin^{2}(x) dx$ is:

$\int_{0}^{\frac{\pi}{2}} \frac{1-\cot x}{\operatorname{cosec} x+\cos x} d x=$

Evaluate the integral: $\int_0^\pi \frac{x}{\sin x}(3 \cos^2 x + 2 \sin x + \sin^3 x - 3) dx$

For any integer $n$, the value of the integral $\int_0^\pi e^{\cos^2 x} \cos^3(2n+1)x \, dx$ is: