Cho hàm số f(x) xác định trên $\Large\mathbb{R} \backslash\left \{ -1;

Cho hàm số f(x) xác định trên $\Large\mathbb{R} \backslash\left \{ -1;1 \right \}$ và thỏa mãn $\Large\mathrm{f^{\prime}(x)=\dfrac{1}{x^{2}-1},f(-3)+f(3)=0}$ và $\Large f\left(-\dfrac{1}{2}\right)+f\left(\dfrac{1}{2}\right)=2 .$ Tính giá trị biểu thức $\Large\mathrm{P=f(0)+f(4)}$

 

• A.

  $\Large\mathrm{P=2+\ln \dfrac{3}{5}}$

• B.

  $\Large\mathrm{P=1+\ln \dfrac{3}{5}}$

• C.

  $\Large\mathrm{P=1+\dfrac{1}{2} \ln \dfrac{3}{5}}$

• D.

  $\Large\mathrm{P=\dfrac{1}{2} \ln \dfrac{3}{5}}$

$\Large\mathrm{f^{\prime}(x)=\dfrac{1}{x^{2}-1} \Rightarrow f(x)=\int \dfrac{d x}{x^{2}-1}=\int \dfrac{d x}{(x-1)(x+1)}$$\Large =\left\{\begin{array}{l}\dfrac{1}{2} \ln \left|\dfrac{x-1}{x+1}\right|+C_{1} \text { khi } x \in(-\infty ;-1) \cup(1 ;+\infty) \\ \dfrac{1}{2} \ln \left|\dfrac{x-1}{x+1}\right|+C_{2} \text { khi } x \in(-1 ; 1)\end{array} \quad\right. }$

Tacó $\Large\mathrm{f(-3)+f(3)=0 \Rightarrow \dfrac{1}{2} \ln 2+C_{1}+\dfrac{1}{2} \ln \dfrac{1}{2}+C_{1}=0 \Leftrightarrow C_{1}=0}$

và $\Large\mathrm{f\left(-\dfrac{1}{2}\right)+f\left(\dfrac{1}{2}\right)=2 \Rightarrow \dfrac{1}{2} \ln 3+C_{2}+\dfrac{1}{2} \ln \dfrac{1}{3}+C_{2}=2 \Leftrightarrow C_{2}=1}$

Suy ra: $\Large\mathrm{f(x)=\left\{\begin{array}{l}\dfrac{1}{2} \ln \left|\dfrac{x-1}{x+1}\right| \text { khi } x \in(-\infty ;-1) \cup(1 ;+\infty) \\ \dfrac{1}{2} \ln \left| \dfrac{x-1}{x+1}\right|+1 \text { khi } x \in(-1 ; 1)\end{array} \Rightarrow P=f(0)+f(4)=1+\dfrac{1}{2} \ln \dfrac{3}{5}\right. }$ đáp án C.