Đáp án:

\( \frac{f'(x)}{f(x)} = \frac{1}{\sqrt{3x+1}} \implies \int \frac{f'(x)}{f(x)} \, dx = \int \frac{1}{\sqrt{3x+1}} \, dx. \)

\(= \frac{1}{3} \int 3(3x+1)^{-\frac{1}{2}} \, dx = \frac{2}{3} \sqrt{3x+1} + C.\)

\( \Rightarrow \ln(f(x))  = \frac{2}{3} \sqrt{3x+1} + C. \)

\( x = 1 \implies  \frac{4}{3}  + C = 0 \implies C = -\frac{4}{3} \)

\(\Rightarrow \ln(f(x)) = \frac{2}{3} \sqrt{3x+1} - \frac{4}{3}. \)

\( \Rightarrow f(x) = e^{\frac{2}{3} \sqrt{3x+1} - \frac{4}{3}}. \)

\(\Rightarrow f(5) \approx 3.79366 \Rightarrow \boxed{C} \)