If $\int \frac{2\sin x}{\ln(1 + (f(x)) + C}$ (where $x > 0$ and $C$ is the constant of integration) then the range of $f(x)$ is
Step-by-Step Solution
Key Concept: Use the substitution $\sqrt{e^x - 1} = t$ to simplify the integral into a standard logarithmic form.
Given $\operatorname{lcx} - 1 = t^2$, we have $e^x dx = 2t dt$. Then $I = \int \frac{2dt}{t} = 2\ln|t| + C = 2\tan^{-1}(\sqrt{e^x - 1}) + C$. For $x \in (0, \infty)$, since $e^x \in (1, \infty)$, we get $f(x) = \sqrt{e^x - 1}$ on $(0, \infty)$.
Correct Answer: A