$\int_0^a f(x) \, dx = $

$\int_0^{\frac{\pi}{2}} \frac{x \tan x \sec^2 x}{\tan^4 x + 1} dx =$

For $0 < a < 1$,the value of the integral $\int_0^\pi \frac{d x}{1-2 a \cos x+a^2}$ is:

Let $f(x)$ be a continuous periodic function with period $T$. Let $I = \int_{a}^{a+T} f(x) \, dx$. Then

The value of the integral $\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \frac{x}{1+\sin x} dx$ is

If $I=\int_0^{\frac{\pi}{2}} \frac{\sin ^{\frac{3}{2}} x}{\sin ^{\frac{3}{2}} x+\cos ^{\frac{3}{2}} x} d x$,then the value of $\int_0^{\frac{\pi}{2}} \frac{x \sin x \cos x}{\sin ^4 x+\cos ^4 x} d x$ is: