$\displaystyle \int\limits_0^1 \frac{5x+8}{x^2+3x+2}   dx$ is

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To solve this problem, we can break apart the fraction using partial fraction decomposition.

$$\frac{5x+8}{x^2+3x+2} = \frac{A}{x+2} + \frac{B}{x+1}$$ $$Ax+A+Bx+2B = 5x+8$$ $$A+B=5 \hskip{2em} A+2B=8$$ $$B=3, A=2$$ $$\frac{5x+8}{x^2+3x+2} = \frac{2}{x+2} + \frac{3}{x+1}$$

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$$\int\limits_0^1\frac{2}{x+2} + \frac{3}{x+1}   dx$$ $$= 2\ln{(x+2)} + 3\ln{(x+1)} \Big|_0^1$$ $$= 2\ln{(1+2)} + 3\ln{(1+1)} - [2\ln{(0+2)} + 3\ln{(0+1)}]$$ $$= 2\ln{3}+3\ln{2}-2\ln{2}-3\ln{1}$$ $$= \ln{9}+\ln{8}-\ln{4}-\ln{1}$$ $$= \ln{\left( \frac{9\cdot8}{4\cdot1}\right)}$$ $$= \boxed{\ln{(18)}}$$