If the tangent to the curve $xy + ax + by = 0$ at $(1,1)$ makes an angle of $\tan^{-1}2$ with positive direction of the $x$-axis, then the value of $\dfrac{ab}{a+b}$ is...

If the side of an equilateral triangle increases at the rate of $\sqrt{3}\ \text{cm/sec}$, then the rate of change of increase of its area when the side is $12\ \text{cm}$ is ____

$\int\sin(\log x)\,dx = $

The value of $\int\dfrac{n\sqrt{\text{cosec}^2 x^n - 1}}{x^{(1-n)}}\,dx$ is ...