Đáp án:

•  \( f(x) = \int \frac{x}{x^2+1} \, dx = \frac{1}{2} \int \frac{2x}{x^2+1} \, dx = \frac{1}{2} \ln |x^2+1| + C \) 

•  \( f(1) = \frac{1}{2} \ln 2 + C = \ln 2 \Leftrightarrow C = \frac{1}{2} \ln 2  \) 

•  \(f(x) = \frac{1}{2} \ln |x^2+1| + \frac{1}{2} \ln 2 \Leftrightarrow 
f(3) = \frac{1}{2} \ln 10 + \frac{1}{2} \ln 2 = \frac{1}{2} \ln 20\)

Vậy chọn đáp án \(\boxed{\text{B}} \)