Solution
= Solution
The <partial fraction decomposition> is
$$
\frac{3x^2+5x+1}{x(x+1)(x+2)}=\frac1{2x}+\frac1{x+1}+\frac3{2(x+2)}.
$$
Integrating between one and two gives
$$
\boxed{\frac12\ln2+\ln\frac32+\frac32\ln\frac43=\frac12\ln\frac{32}{3}.}
$$
All logarithm arguments are positive on this integration interval.