​​​​​​​Đáp án:

\( 3 f(x) +f(x)+ x f'(x) = x^{2017} \)

\( \Leftrightarrow 3 f(x) + \left( x f(x) \right)' = x^{2017} \)

\( \Leftrightarrow 3 \int_{0}^{1} f(x) dx + x f(x) \big|_{0}^{1} = \int_{0}^{1} x^{2017} dx \)

\( \Rightarrow 3 \int_{0}^{1} f(x) dx + f(1) = \frac{x^{2018}}{2018} \Big|_{0}^{1} \)

\( f(1) = \frac{1}{2021} \Rightarrow I = \text{A} \)

\( 4 f(x) + x f'(x) = x^{2017} \) (Nhân cả hai vế với \( x^3 \))

\( \Rightarrow 4 x^3 f(x) + x^4 f'(x) = x^{2020} \)

\( \Rightarrow (x^4 f(x))' = x^{2020} \)

\( \Rightarrow x^4 f(x) = \frac{x^{2021}}{2021} + C \)

\( f(1) = \frac{1}{2021}  \Rightarrow C = 0 \)

\( \Rightarrow f(x) = \frac{x^{2017}}{2021} \Rightarrow  \int_{0}^{1} f(x) dx = \frac{1}{2018 \cdot 2021}\Rightarrow \boxed{A}\)