Cho hàm số \(f\left( x \right)\) xác định trên \(R...

D

- Hướng dẫn giải

Phương pháp giải:

\(f\left( x \right)=\int\limits_{{}}^{{}}{f'\left( x \right)dx}\)

Giải chi tiết:

\(\begin{array}{l}
\,\,\,\,f\left( x \right) = \int\limits_{}^{} {f'\left( x \right)dx} = \int\limits_{}^{} {\frac{1}{{{x^2} - 1}}dx} = \frac{1}{2}\ln \left| {\frac{{x - 1}}{{x + 1}}} \right| + C\\
\Rightarrow f\left( x \right) = \left[ \begin{array}{l}
\frac{1}{2}\ln \frac{{x - 1}}{{x + 1}} + {C_1}\,\,\,\,\,\,\,khi\,x \in \left( { - \infty ; - 1} \right) \cup \left( {1; + \infty } \right)\\
\frac{1}{2}\ln \frac{{1 - x}}{{x + 1}} + {C_2}\,\,\,\,\,\,\,khi\,x \in \left( { - 1;1} \right)
\end{array} \right.\\
\Rightarrow f\left( { - 3} \right) + f\left( 3 \right) = \frac{1}{2}\ln 2 + {C_1} + \frac{1}{2}\ln \frac{1}{2} + {C_1} = 0 \Leftrightarrow {C_1} = 0\\
\,\,\,\,\,\,f\left( { - \frac{1}{2}} \right) + f\left( 3 \right) = \frac{1}{2}\ln 3 + {C_2} + \frac{1}{2}\ln \frac{1}{3} + {C_2} = 2 \Leftrightarrow {C_2} = 1\\
\Rightarrow f\left( x \right) = \left[ \begin{array}{l}
\frac{1}{2}\ln \frac{{x - 1}}{{x + 1}}\,\,\,\,\,\,\,\,\,\,\,\,khi\,x \in \left( { - \infty ; - 1} \right) \cup \left( {1; + \infty } \right)\\
\frac{1}{2}\ln \frac{{1 - x}}{{x + 1}}\, + 1\,\,\,\,\,khi\,x \in \left( { - 1;1} \right)
\end{array} \right.\\
\Rightarrow f\left( { - 2} \right) + f\left( 0 \right) + f\left( 4 \right) = \frac{1}{2}\ln 3 + \frac{1}{2}\ln 1 + 1 + \frac{1}{2}\ln \frac{3}{5} = 1 + \frac{1}{2}\ln \frac{9}{5}
\end{array}\)

Chọn D.