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Câu hỏi: Cho hàm số $f\left( x \right)$ xác định trên $\mathbb{R}\backslash \left\{ \dfrac{1}{2} \right\}$ thỏa mãn ${f}'\left( x \right)=\dfrac{2}{2x-1}$ và $f\left( 0 \right)=1; f\left( 1 \right)=-2$. Giá trị của biểu thức $f\left( -1 \right)+f\left( 3 \right)$ bằng
A. $2+\ln 15$.
B. $3-\ln 15$.
C. $\ln 15-1$.
D. $\ln 15$.
A. $2+\ln 15$.
B. $3-\ln 15$.
C. $\ln 15-1$.
D. $\ln 15$.
$f\left( x \right)=\int{{f}'\left( x \right)dx=\int{\dfrac{2}{2x-1}dx=\int{\dfrac{2.\dfrac{1}{2}d\left( 2x-1 \right)}{2x-1}}}}=\ln \left| 2x-1 \right|+c=\left\{ \begin{aligned}
& \ln \left( 2x-1 \right)+{{C}_{1}} \text{khi} x>\dfrac{1}{2} \\
& \ln \left( 1-2x \right)+{{C}_{2}} \text{khi} x<\dfrac{1}{2} \\
\end{aligned} \right.$.
$f\left( 1 \right)=-2\Leftrightarrow {{C}_{1}}=-2$ $\Rightarrow f\left( x \right)=\ln \left( 2x-1 \right)-2$
$f\left( 0 \right)=1$ $\Leftrightarrow {{C}_{2}}=1$ $\Rightarrow f\left( x \right)=\ln \left| 2x-1 \right|+1$.
$\Rightarrow $ $\left\{ \begin{aligned}
& f\left( -1 \right)=\ln 3+1 \\
& f\left( 3 \right)=\ln 5-2 \\
\end{aligned} \right. $ $ \Leftrightarrow f\left( -1 \right)+f\left( 3 \right)=\ln 15-1$.
& \ln \left( 2x-1 \right)+{{C}_{1}} \text{khi} x>\dfrac{1}{2} \\
& \ln \left( 1-2x \right)+{{C}_{2}} \text{khi} x<\dfrac{1}{2} \\
\end{aligned} \right.$.
$f\left( 1 \right)=-2\Leftrightarrow {{C}_{1}}=-2$ $\Rightarrow f\left( x \right)=\ln \left( 2x-1 \right)-2$
$f\left( 0 \right)=1$ $\Leftrightarrow {{C}_{2}}=1$ $\Rightarrow f\left( x \right)=\ln \left| 2x-1 \right|+1$.
$\Rightarrow $ $\left\{ \begin{aligned}
& f\left( -1 \right)=\ln 3+1 \\
& f\left( 3 \right)=\ln 5-2 \\
\end{aligned} \right. $ $ \Leftrightarrow f\left( -1 \right)+f\left( 3 \right)=\ln 15-1$.
Đáp án C.