$\int\limits_0^\infty [2e^{-x}]\, dx$,where $[x]$ denotes the greatest integer function,is equal to:

Let $[x]$ denote the greatest integer $\leq x$. Consider the function $f(x) = \max \{x^2, 1 + [x]\}$. Then the value of the integral $\int_0^2 f(x) dx$ is:

If the value of the integral $\int_{0}^{1/2} \frac{x^{2}}{(1-x^{2})^{3/2}} dx$ is $\frac{k}{6}$,then $k$ is equal to

$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin (x-[x]) \, dx=$

If $[\bullet]$ denotes the greatest integer function, then evaluate $\int_0^{2 \pi} [|\sin x| + |\cos x|] \, dx$.

Let $f$ be a continuous function in $[0, 1]$, then $\lim_{n \rightarrow \infty} \sum_{j=0}^n \frac{1}{n} f\left(\frac{j}{n}\right)$ is