The number of continuous functions $f : [0, \frac{3}{2}] \rightarrow (0, \infty)$ satisfying the equation $4 \int_0^{3/2} f(x) dx + 125 \int_0^{3/2} \frac{dx}{\sqrt{f(x)+x^2}} = 108$ is

Given that for each $a \in (0,1)$,the limit $g(a) = \lim_{n \rightarrow 0^{+}} \int_n^{1-n} t^{-a}(1-t)^{a-1} dt$ exists. In addition,it is given that the function $g(a)$ is differentiable on $(0,1)$.
$1.$ The value of $g\left(\frac{1}{2}\right)$ is
$(A) \pi$ $(B) 2\pi$ $(C) \frac{\pi}{2}$ $(D) \frac{\pi}{4}$
$2.$ The value of $g'\left(\frac{1}{2}\right)$ is
$(A) \frac{\pi}{2}$ $(B) \pi$ $(C) -\frac{\pi}{2}$ $(D) 0$
Select the correct pair of answers for $1$ and $2$.

Let $f(\theta) = \sin \theta + \int_{-\pi / 2}^{\pi / 2} (\sin \theta + t \cos \theta) f(t) dt$. Then the value of $\left| \int_{0}^{\pi / 2} f(\theta) d\theta \right|$ is

If $f(x) = A \sin \left( \frac{\pi x}{2} \right) + B$,$f'(1/2) = \sqrt{2}$ and $\int_{0}^{1} f(x) dx = \frac{2A}{\pi}$,then the constants $A$ and $B$ are respectively.

For each real number $x$,let $[x]$ denote the greatest integer less than or equal to $x$,and let $\{x\} = x - [x]$. Then the smallest positive integer $M$ for which $\int_1^M \{x\}^{[x]} dx > 1$ is

If $\alpha = \int_{0}^{2\sqrt{3}} \log_2(x^2+4) dx + \int_{2}^{4} \sqrt{2^x-4} dx$, then $\alpha^2$ is equal to . . . . . . .