Let a differentiable function $f$ satisfy $f(x) + \int_{3}^{x} \frac{f(t)}{t} dt = \sqrt{x+1}$ for $x \geq 3$. Then $12f(8)$ is equal to:

Let $g$ be a differentiable function such that $\int_0^x g(t) dt = x - \int_0^x tg(t) dt$ for $x \geq 0$. Let $y = y(x)$ satisfy the differential equation $\frac{dy}{dx} - y \tan x = 2(x+1) \sec x g(x)$ for $x \in [0, \frac{\pi}{2})$. If $y(0) = 0$,then $y(\frac{\pi}{3})$ is equal to

Let a curve $y = y(x)$ pass through the point $(3,3)$ and the area of the region under this curve,above the $x$-axis and between the abscissae $3$ and $x (>3)$ be $\left(\frac{y}{x}\right)^{3}$. If this curve also passes through the point $(\alpha, 6\sqrt{10})$ in the first quadrant,then $\alpha$ is equal to $........$

Let $f: [1, \infty) \to \mathbb{R}$ be a differentiable function defined as $f(x) = \int_1^x f(t) \, dt + (1 - x)(\log_e x - 1) + e$. Then the value of $f(f(1))$ is:

Let $y=y(x)$ be the solution of the differential equation $(xy-5x^2\sqrt{1+x^2})dx+(1+x^2)dy=0$,with $y(0)=0$. Then $y(\sqrt{3})$ is equal to

Let $f: R \rightarrow R$ be a continuous function which satisfies $f(x) = \int_{0}^{x} f(t) \, dt$. Then, the value of $f(\log_{e} 5)$ is: