The value of $\int_{-1}^{2} \frac{|x|}{x} d x$ is

Let ${I_1} = \int_1^2 \frac{dx}{\sqrt{1 + x^2}}$ and ${I_2} = \int_1^2 \frac{dx}{x}$,then:

$\int_0^{32 \pi} \sqrt{1-\cos 4 x} \, dx =$ (in $\sqrt{2}$)

If $[x]$ denotes the greatest integer less than or equal to $x$,then the value of $\int_{1}^{5} [|x - 3|] \, dx$ is

Evaluate the definite integral $\int_{0}^{\frac{\pi}{2}} \cos ^{2} x \,d x$.

Suppose $g(x) = \int_0^x f(t) dt$,where $f$ is such that for $t \in [0, 1]$,$0 \le f(t) \le \frac{1}{2}$ and for $t \in [1, 2]$,$\frac{1}{2} \le f(t) \le 1$. Find the range of $g(2)$.