Hướng dẫn giải

Ta có $\int\limits_{ - 1}^0 {f'\left( x \right)dx} = f\left( 0 \right) - f\left( { - 1} \right)$ nên suy ra $f\left( { - 1} \right) = f\left( 0 \right) - \int\limits_{ - 1}^0 {f'\left( x \right)dx.}$

$= 1 - \int\limits_{ - 1}^0 {f'\left( x \right)dx.}$

Tương tự ta cũng có

$f\left( 3 \right) = f\left( 1 \right) + \int\limits_1^3 {f'\left( x \right)dx}$

$= - 2 + \int\limits_1^3 {f'\left( x \right)dx}$.

Vậy $f\left( { - 1} \right) + f\left( 3 \right) = - 1 - \int\limits_{ - 1}^0 {f'\left( x \right)dx} + \int\limits_1^3 {f'\left( x \right)dx} = - 1 - \ln \left| {2x - 1} \right|\left| {_{\scriptstyle\atop\scriptstyle - 1}^{\scriptstyle0\atop\scriptstyle}} \right. + \ln \left| {2x - 1} \right|\left| {_{\scriptstyle\atop\scriptstyle1}^{\scriptstyle3\atop\scriptstyle}} \right..$

Vậy $f\left( { - 1} \right) + f\left( 3 \right) = - 1 + \ln 15.$

Chọn A.