Hướng dẫn giải

Ta có $\int\limits_1^2 {\frac{{5x + 6}}{{{x^2} + 5x + 6}}dx = \int\limits_1^2 {\frac{{5x + 6}}{{\left( {x + 2} \right)\left( {x + 3} \right)}}dx} = \int\limits_1^2 {\left( {\frac{9}{{x + 3}} - \frac{4}{{x + 2}}} \right)dx} }$

                                    $= \left( {9\ln \left| {x + 3} \right| - 4\ln \left| {x + 2} \right|} \right)\left| {_{\scriptstyle\atop\scriptstyle1}^{\scriptstyle2\atop\scriptstyle}} \right. = 9\ln 5 + 4\ln 3 - 26\ln 2.$

Suy ra $a = - 26,b = 4,c = 9.$ Vậy $S = a + bc = - 26 + 4.9 = 10.$

Chọn B.