Using the Trapezoidal rule, find the approximate value of $\int_1^4 y \, dx$ based on the following data:

$x$	$1$	$2$	$3$	$4$
$y$	$0.7111$	$0.7222$	$0.7333$	$0.7444$

(in $.1833$)

The area of the region under the graph of $y = xe^{-ax}$ as $x$ varies from $0$ to $\infty$,where $'a'$ is a positive constant,is

$\int_0^{2\pi} e^{x/2} \sin \left( \frac{x}{2} + \frac{\pi}{4} \right) \, dx = $

If $[x]$ denotes the greatest integer function of $x$ and $\int_{-\frac{3}{2}}^{\frac{3}{2}}[2x-3] dx = k$, then $\left|k+\frac{1}{2}\right| = $

Let $f: R \rightarrow R$ be defined as $f(x)=e^{-x} \sin x$. If $F :[0,1] \rightarrow R$ is a differentiable function such that $F(x)=\int_{0}^{x} f(t) dt$,then the value of $\int_{0}^{1}(F'(x)+f(x)) e^{x} dx$ lies in the interval

$\int_0^\pi \frac{dx}{1 + \sin x} = $