Evaluate the definite integral $\int_{4}^{5} e^{x} \, dx$.

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 $.1833$)

$\int_{0}^{\frac{\pi}{2}} \sin^{2} x \, dx =$

If $I_n = \int_0^{\frac{\pi}{2}} \cos^n x \cos(nx) dx$, then $I_1, I_2, I_3, \ldots$ are in

$\int_0^1 \log (x+1) \, dx =$

Let $n$ be a positive integer. For a real number $x$,let $[x]$ denote the largest integer not exceeding $x$ and $\{x\}=x-[x]$. Then,$\int \limits_1^{n+1} \frac{(\{x\})^{[x]}}{[x]} d x$ is equal to