Let $f$ be a real-valued function defined on the interval $(0,\,\infty)$ by $f(x)=\ln x+\int_{0}^{x}\sqrt{1+\sin t}\,dt$. Then which of the following statement(s) is (are) true?

$f(x) = \ln x + \int_{0}^{x} \sqrt{1+\sin t}\,dt, \qquad x \in (0,\infty)$ $\text{If } g(x) = \int_0^x h(t)\,dt \text{ then } g'(x) = h(x)$ $\text{Check existence/continuity of } f', f''; \text{ and bounds on } |f|, |f'|$ $f'(x) = \dfrac{1}{x} + \sqrt{1+\sin x}$ $1+\sin x = \left(\sin\tfrac{x}{2}+\cos\tfrac{x}{2}\right)^{2}$ $\sqrt{1+\sin x} = \left|\sin\tfrac{x}{2}+\cos\tfrac{x}{2}\right|$ $\text{corner where } 1+\sin x = 0,\ \text{i.e. } x = \tfrac{3\pi}{2}$ $f' \text{ continuous but not differentiable at } \tfrac{3\pi}{2}$ $\Rightarrow (1) \text{ FALSE}, \qquad (2) \text{ TRUE}$ $|f'(x)| = \left|\dfrac{1}{x}+\sqrt{1+\sin x}\right| \leq 1 + \sqrt{2} \quad (x \geq 1)$ $f(x) \to \infty \ (x \to \infty), \quad f' \text{ bounded} \Rightarrow |f'(x)| < |f(x)| \text{ for large } x$ $\Rightarrow (3) \text{ TRUE}$ $f(x) \to \infty \Rightarrow \text{no } \beta \text{ bounds } |f(x)|+|f'(x)| \Rightarrow (4) \text{ FALSE}$ $\boxed{\text{TRUE statements: } (2) \text{ and } (3)}$ $f'(x) = \tfrac{1}{x} + \sqrt{1+\sin x}; \quad \text{TRUE: } (2), (3)$