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

The value of $\int_{0}^{\pi /2} \frac{\sin^{2/3} x}{\sin^{2/3} x + \cos^{2/3} x} dx$ is

If $I_n = \int_{-\pi}^{\pi} \frac{\sin(nx)}{(1+\pi^x) \sin x} dx$,$n=0, 1, 2, \ldots$,then
$(A)$ $I_n = I_{n+2}$
$(B)$ $\sum_{m=1}^{10} I_{2m+1} = 10\pi$
$(C)$ $\sum_{m=1}^{10} I_{2m} = 0$
$(D)$ $I_n = I_{n+1}$

$\int\limits_0^{\frac{\pi }{2}} {\sqrt {\sin 2\theta } } \sin \theta \,d\theta$ is equal to :

The value of $\int_{-7}^{7} \frac{5^x}{5^{[x]}} dx$ is equal to (where $[.]$ denotes the greatest integer function).

$\int_0^{\pi /2} \frac{x \sin x \cos x}{\cos^4 x + \sin^4 x} \, dx = $