If the area bounded by $y = x^3 + ax$ (where $a > 0$), the $x$-axis and the lines $x = -2$ and $x = 1$ is $\dfrac{37}{4}$ square units, then $\ldots$

The rate of disintegration of a radioactive element at any time is proportional to its mass at that time, where $k$ $(k>0)$ is the constant of proportionality. The time during which an original mass of 1.5 gm will disintegrate to a mass of 0.5 gm is $\ldots$

The differential equation of the family of all parabolas whose axis is the $y$-axis is $\ldots$

The integrating factor of the differential equation $x\dfrac{dy}{dx} + 2y = x^2 \log x$ is