Cho hàm số f(x) xác định trên R\{-1;1} và thỏa mãn

* Hướng dẫn giải

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) =\int \mathrm{f}\text{'}\left(\mathrm{x}\right) =\int \frac{\mathrm{dx}}{{\mathrm{x}}^2-1}=\frac{1}{2}\ln \left|\frac{\mathrm{x}-1}{\mathrm{x}+1}\right|+\mathrm{C} \\ =\left\{\begin{array}{l}\frac{1}{2}\ln \frac{\mathrm{x}-1}{\mathrm{x}+1}+{\mathrm{C}}_1 \mathrm{khi} \mathrm{x}>1\\ \frac{1}{2}\ln \frac{1-\mathrm{x}}{\mathrm{x}+1}+{\mathrm{C}}_2 \mathrm{khi} -1<\mathrm{x}<1\\ \frac{1}{2}\ln \frac{\mathrm{x}-1}{\mathrm{x}+1}+{\mathrm{C}}_3 \mathrm{khi} \mathrm{x}<-1\end{array}\right. \\ \mathrm{Ta} \mathrm{có} f\left(-3\right)+f\left(3\right)=0\iff \frac{1}{2}\ln 2+\frac{1}{2}\ln \frac{1}{2}+{\mathrm{C}}_3+{\mathrm{C}}_1=0 \\ \iff {\mathrm{C}}_3+{\mathrm{C}}_1=0 \\ \mathrm{Và} \mathrm{f} \left(-\frac{1}{2}\right)+\mathrm{f} \left(\frac{1}{2}\right)=2\iff \frac{1}{2}\ln 3+{\mathrm{C}}_2+\frac{1}{2}\ln \frac{1}{3}+{\mathrm{C}}_2=2 \end{gathered}$$