Tìm giá trị của a để I = intlimits_0a frac5x + 7x2 + 3x + 2dx = 3ln 2 + 2ln 3.

$\begin{array}{l} I = \int\limits_0^a {\frac{{5x + 7}}{{{x^2} + 3x + 2}}dx} = \int\limits_0^a {\frac{{5\left( {x + 1} \right) + 2}}{{\left( {x + 1} \right)\left( {x + 2} \right)}}} dx = \int\limits_0^a {\frac{5}{{x + 2}}dx + \int\limits_0^a {\frac{2}{{x + 1}}dx} - \int\limits_0^a {\frac{2}{{x + 2}}dx} } \\ = \left. {5\ln \left| {x + 2} \right|} \right|_0^a + \left. {2\ln \left| {x + 1} \right|} \right|_0^a - \left. {2\ln \left| {x + 2} \right|} \right|_0^a = \left. {3\ln \left( {x + 2} \right)} \right|_0^a + \left. {2\ln \left| {x + 1} \right|} \right|_0^a\\ = 3\ln \left| {a + 2} \right| - 3\ln 2 + 2\ln \left| {a + 1} \right| = 3\ln \left| {\frac{{a + 2}}{2}} \right| + 2\ln \left| {a + 1} \right| = 3\ln 2 + 2\ln 3 \Leftrightarrow a = 2. \end{array}$