\int_2^3\frac{1-2x}{5-x}\,\mathrm{d}x=\int_2^3\Bigl(2-\frac{9}{5-x}\Bigr)\,\mathrm{d}x=2x\Big|_2^3+9\ln|5-x|\Big|_2^3=2+[9\ln (5-3)-9\ln (5-2)]=2-9\ln\frac{3}{2}