In the given figure,each box represents a function machine. $A$ function machine illustrates what it does with the input. Which of the following statements is correct?

$\int \frac{3}{(2-x)^2} \, dx$ is equal to $:-$

If $log_{10} (xy) = 2$,then the value of $xy$ is

If $x^3+3xy+y^3=1$,then the correct option$(s)$ is/are:-
$(A) \left(\frac{dy}{dx}\right)_{(1,1)}=-1$
$(B) \left(\frac{dy}{dx}\right)_{(1,1)}=-2$
$(C) \left(\frac{dy}{dx}\right)_{(1,0)}=-1$
$(D) \left(\frac{dy}{dx}\right)_{(1,0)}=-3$

$\int \frac{dx}{{(2ax - x^2)}^{1/2}} = a^n \sin^{-1} \left( \frac{x}{a} - 1 \right)$. In this formula,$n =$ . . . . . . .

${d \over {dx}}({x^2}{e^x}\sin x) = $