Let $f : R \to R$ be a function such that $f(2 - x) = f(2 + x)$ and $f(4 - x) = f(4 + x)$,for all $x \in R$. If $\int_{0}^{2} f(x) dx = 5$,then the value of $\int_{10}^{50} f(x) dx$ is:

If $\int_{0}^{100 \pi} \frac{\sin ^{2} x}{e^{\left(\frac{x}{\pi}-\left[\frac{x}{\pi}\right]\right)}} d x=\frac{\alpha \pi^{3}}{1+4 \pi^{2}}, \alpha \in R$,where $[x]$ is the greatest integer less than or equal to $x$,then the value of $\alpha$ is :

Let $I = \int_{\pi / 4}^{\pi / 3} \frac{\sin x}{x} dx$. Then

Given $\int_{0}^{\frac{\pi}{2}} \frac{dx}{1 + \sin x + \cos x} = \ln 2$,then the value of the definite integral $\int_{0}^{\frac{\pi}{2}} \frac{\sin x}{1 + \sin x + \cos x} dx$ is equal to

If $f(x) = \int\limits_1^x \frac{\ln t}{1 + t} dt$ where $x > 0$,then the value$(s)$ of $x$ satisfying the equation $f(x) + f(1/x) = 0$ is/are:

$\int_{\pi / 11}^{9 \pi / 22} \frac{d x}{1+\sqrt{\tan x}} = $