If $f(x)$ is a continuous periodic function with period $T,$ then the integral $I = \int_a^{a + T} {f(x)\,dx} $ is

If ${I_n} = \int_0^{\pi /4} {{\tan ^n}\theta \,d\theta }$,then ${I_8} + {I_6}$ equals

Let $f:(0,2) \rightarrow R$ be defined as $f(x) = \log_{2}\left(1+\tan\left(\frac{\pi x}{4}\right)\right)$. Then,$\lim_{n \rightarrow \infty} \frac{2}{n}\left(f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)+\ldots+f(1)\right)$ is equal to

Let $f : R \rightarrow R$ be a continuous function satisfying $f(x) + f(x + k) = n$,for all $x \in R$,where $k > 0$ and $n$ is a positive integer. If $I_{1} = \int_{0}^{4nk} f(x) dx$ and $I_{2} = \int_{-k}^{3k} f(x) dx$,then:

If $\int_{ - a}^a {\sqrt {\frac{{a - x}}{{a + x}}} \,dx = k\pi ,} $ then $k = $

Consider $L = \sqrt[3]{2012} + \sqrt[3]{2013} + \ldots + \sqrt[3]{3011}$,$R = \sqrt[3]{2013} + \sqrt[3]{2014} + \ldots + \sqrt[3]{3012}$,and $I = \int_{2012}^{3012} \sqrt[3]{x} \, dx$. Then,