\(\begin{array}{l}
\quad I = \displaystyle\int\limits_1^2\dfrac{x^3 - 1}{x^2 + x}dx\\
\Leftrightarrow I = \displaystyle\int\limits_1^2\left(x + \dfrac{2}{x+1} - \dfrac{1}{x} - 1\right)dx\\
\Leftrightarrow I = \left(\dfrac{x^2}{2} + 2\ln|x+1| - \ln|x| - x\right)\Bigg|_1^2\\
\Leftrightarrow I = \left(2 + 2\ln3 - \ln2 - 2\right) - \left(\dfrac12 + 2\ln2 - \ln1 - 1\right)\\
\Leftrightarrow I = \dfrac12 + 2\ln3 - 3\ln2\\
\Rightarrow \begin{cases}a = \dfrac12\\b = 2\\c = -3\end{cases}\\
\end{array}\)
