Lời giải

•  \( \frac{f'(x)}{f(x)} = 2 \implies \int \frac{f'(x)}{f(x)} \, dx = \int 2 \, dx \implies \ln (f(x)) = 2x + C\)

\( \Rightarrow f(x) = e^{2x + C}  \)  

\( f(0) = 1 \Rightarrow e^C= 1 \Leftrightarrow C = 0 \)

\( \Rightarrow f(x) = e^{2x} \Rightarrow f(1) = e^2 \)

Do đó: \( I = \frac{1}{2}(e^2 - 1)\)

\(\Rightarrow\) Vậy chọn đáp án \(\boxed{\text{B}} \)

Cách 2: 

\( \frac{f'(x)}{f(x)} = 2 \Rightarrow f(x) = \frac{1}{2} f'(x) \Rightarrow \int_0^1 f(x) \, dx = \frac{1}{2} \int_0^1 f'(x) \, dx = \frac{1}{2} (f(1) - f(0)) \Rightarrow f(x) = e^{2x}
\)

\(I = \int_0^1 f(x) \, dx = \int_0^1 e^{2x} \, dx = \frac{1}{2} e^{2x} \Big|_0^1 = \frac{1}{2} \left(e^2 - 1\right)\)