Đáp án:
\[a + b + c = 0\]
Giải thích các bước giải:
\(\begin{array}{l}
t = 2 + {x^2} \Rightarrow \left\{ \begin{array}{l}
dt = 2xdx\\
x = 0 \Rightarrow t = 2\\
x = 1 \Rightarrow t = 3
\end{array} \right.\\
\int\limits_0^1 {x.\ln \left( {2 + {x^2}} \right)dx} = \int\limits_2^3 {\ln t.\frac{{dt}}{2}} = \frac{1}{2}\int\limits_2^3 {\ln tdt} \\
\left\{ \begin{array}{l}
u = \ln t\\
v' = 1
\end{array} \right. \Rightarrow \left\{ \begin{array}{l}
u' = \frac{1}{t}\\
v = t
\end{array} \right.\\
\Rightarrow \frac{1}{2}\int\limits_2^3 {\ln tdt} = \frac{1}{2}.\left[ {\mathop {\left. {t.\ln t} \right|}\nolimits_2^3 - \int\limits_2^3 {\frac{1}{t}.tdt} } \right] = \frac{1}{2}.\left[ {3\ln 3 - 2\ln 2 - 1} \right]\\
\Rightarrow a = \frac{3}{2};\,\,b = - 1;\,\,c = - \frac{1}{2}\\
\Rightarrow a + b + c = 0
\end{array}\)
