If $y = \sqrt{\cos x^2 + \sqrt{\cos x^2 + \sqrt{\cos x^2 + \ldots \infty}}}$ and $\dfrac{dy}{dx} = \dfrac{f(x)}{2y - 1}$ then, $\int f(x)\, dx = \ldots$

If $f(x) = \cot^{-1}\left(\dfrac{3x - x^3}{1 - 3x^2}\right)$ and $g(x) = \cos^{-1}\left(\dfrac{1 - x^2}{1 + x^2}\right)$, then $\lim\limits_{x \to a}\dfrac{f(x) - f(a)}{g(x) - g(a)}$, $\left(0 < a < \dfrac{1}{2}\right)$ is

If $y = x \cdot 7^x$, then the value of $\dfrac{dx}{dy}$ when $x = 1$ is

If $y = [(x+1)(2x+1)(3x+1)\ldots\ldots(nx+1)]^4$, where $n \in N$ and $\dfrac{dy}{dx}$ at $x = 0$ is $2k$, then the value of $k$ is