Evaluate the definite integral $\int_0^2 [2x] \, dx$,where $[.]$ denotes the greatest integer function.

For a real number $x$,let $[x]$ denote the largest integer less than or equal to $x$ and $\{x\}=x-[x]$. The smallest possible integer value of $n$ for which $\int_1^n [x]\{x\} dx$ exceeds $2013$ is

Evaluate the integral: $\int_0^{\pi /2} \frac{1 + 2\cos x}{(2 + \cos x)^2} dx$

If $2f(x) - 3f\left( \frac{1}{x} \right) = x$,then $\int_1^2 f(x) \, dx$ is equal to

If $a_n = \int_0^{\frac{\pi}{2}} \frac{\sin^2 nx}{\sin x} dx$,then $a_2 - a_1, a_3 - a_2, a_4 - a_3$ are in

Let $f(x) = \begin{cases} \frac{x}{\sin x}, & x \in (0, 1) \\ 1, & x = 0 \end{cases}$. Consider the integral $I_n = \sqrt{n} \int_0^{1/n} f(x) e^{-nx} dx$. Then,$\lim_{n \to \infty} I_n$ is: