If $f(x) = \int_{-1}^{x} |t| dt$,then for any $x \geq 0$,$f(x)$ 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:

Let $I = \int_{0}^{1} \frac{\sin x}{\sqrt{x}} \, dx$ and $J = \int_{0}^{1} \frac{\cos x}{\sqrt{x}} \, dx$. Then which one of the following is true?

Let $[x]$ and $\{x\}$ be the integer part and fractional part of a real number $x$ respectively. The value of the integral $\int_0^5 [x]\{x\} dx$ is

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 $.1833$)

If $I$ is the greatest of $I_1=\int_0^1 e^{-x} \cos ^2 x \, dx, I_2=\int_0^1 e^{-x^2} \cos ^2 x \, dx, I_3=\int_0^1 e^{-x^2} \, dx, I_4=\int_0^1 e^{-x^2 / 2} \, dx$, then