\(\int\limits^4_3\dfrac{dx}{x^2+x}=\int\limits^4_3\dfrac{dx}{x\left(x+1\right)}=\int\limits^4_3\left(\dfrac{1}{x}-\dfrac{1}{x+1}\right)dx=\left[lnx-ln\left(x+1\right)\right]|^4_3\)
\(=ln4-ln5-\left(ln3-ln4\right)=2ln4-ln3-ln5\)
\(=4ln2-ln3-ln5\Rightarrow a=4;b=c=-1\)
\(\Rightarrow S=2\)