Let $g(x)$ be an antiderivative for $f(x)$. Then $\ln(1 + (g(x))^2)$ is an antiderivative for

$A$ point $P$ moves in a counter-clockwise direction on a circular path as shown in the figure. The movement of $P$ is such that it sweeps out a length $S = t^3 + 5$,where $S$ is in meters and $t$ is in seconds. The radius of the path is $20 \, m$. The acceleration of $P$ when $t = 2 \, s$ is nearly ......... $m/s^2$.

Let $f(x) = \frac{2\sin^2 x - 1}{\cos x} + \frac{\cos x(2\sin x + 1)}{1 + \sin x}$. Then find $\int e^x (f(x) + f'(x)) dx$ (where $c$ is the constant of integration).

Drift velocity ${v_d}$ varies with the intensity of the electric field $E$ as per the relation:

An object of mass $2m$ is projected with a speed of $100 \ m/s$ at an angle $\theta = \sin^{-1}(3/5)$ to the horizontal. At the highest point,the object breaks into two pieces of same mass $m$ and the first one comes to rest. The distance between the point of projection and the point of landing of the bigger piece (in metre) is (Given,$g = 10 \ m/s^2$)

When a capacitor is connected in series to an $LR$ circuit,the alternating current flowing in the circuit