\( \displaystyle \int\limits_0^1 \frac{5x+8}{x^2+3x+2} dx \) is
To solve this problem, we can break apart the fraction using partial fraction decomposition.
$$ \frac{5x+8}{x^2+3x+2} = \frac{A}{x+2} + \frac{B}{x+1} $$
$$ Ax+A+Bx+2B = 5x+8 $$
$$ A+B=5 \hskip{2em} A+2B=8 $$
$$ B=3, A=2 $$
$$ \frac{5x+8}{x^2+3x+2} = \frac{2}{x+2} + \frac{3}{x+1} $$
$$ \int\limits_0^1\frac{2}{x+2} + \frac{3}{x+1} dx $$
$$ = 2\ln{(x+2)} + 3\ln{(x+1)} \Big|_0^1$$
$$ = 2\ln{(1+2)} + 3\ln{(1+1)} - [2\ln{(0+2)} + 3\ln{(0+1)}] $$
$$ = 2\ln{3}+3\ln{2}-2\ln{2}-3\ln{1} $$
$$ = \ln{9}+\ln{8}-\ln{4}-\ln{1} $$
$$ = \ln{\left( \frac{9\cdot8}{4\cdot1}\right)} $$
$$ = \boxed{\ln{(18)}} $$