$\int_{\pi /4}^{\pi /2} \csc^2 x \, dx = $

$A$ minimum value of $\int_0^x t e^{t^2} d t$ is

Let $f(x) = x - [x],$ for every real number $x,$ where $[x]$ is the integral part of $x.$ Then $\int_{-1}^{1} f(x) \, dx =$

Let for $x \in R$,$S_0(x) = x$,$S_k(x) = C_k x + k \int_0^x S_{k-1}(t) dt$,where $C_0 = 1$,$C_k = 1 - \int_0^1 S_{k-1}(x) dx$,$k = 1, 2, 3, \ldots$. Then $S_2(3) + 6C_3$ is equal to $...........$.

If $[2,6]$ is divided into four intervals of equal length,then the approximate value of $\int_2^6 \frac{1}{x^2-x} dx$ using Simpson's rule is

The value of $\int_{\pi}^{2\pi} [2\sin x] \, dx$,where $[\cdot]$ represents the greatest integer function,is