Evaluate $\int_0^{2 \pi} \cos m x \cos n x \, dx + \int_{-\pi}^\pi \sin m x \cos n x \, dx$ for $m, n \in \mathbb{Z}$.

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

$\int_{0}^{\pi} \frac{x \, dx}{1+\cos \alpha \sin x}, (0 < \alpha < \pi)$ is equal to

Let $T > 0$ be a fixed number. $f: R \rightarrow R$ is a continuous function such that $f(x+T) = f(x)$ for all $x \in R$. If $I = \int_0^T f(x) dx$,then find the value of $\int_0^{5T} f(2x) dx$.

$\int_0^{\pi /4} {\log (1 + \tan \theta )\,d\theta = } $

The value of $\int_0^{\pi / 2} \frac{(\cos x)^{\sin x}}{(\cos x)^{\sin x}+(\sin x)^{\cos x}} d x$ is