Evaluate the definite integral $\int_{2}^{3} \frac{x}{x^{2}+1} dx$.

(N/A) Let $I = \int_{2}^{3} \frac{x}{x^{2}+1} dx$.
To evaluate this,we use the substitution method. Let $u = x^{2} + 1$. Then $du = 2x dx$,which implies $x dx = \frac{1}{2} du$.
Changing the limits of integration:
When $x = 2$,$u = 2^{2} + 1 = 5$.
When $x = 3$,$u = 3^{2} + 1 = 10$.
Substituting these into the integral:
$I = \int_{5}^{10} \frac{1}{2u} du = \frac{1}{2} [\ln |u|]_{5}^{10}$.
Applying the fundamental theorem of calculus:
$I = \frac{1}{2} (\ln 10 - \ln 5) = \frac{1}{2} \ln \left(\frac{10}{5}\right) = \frac{1}{2} \ln 2$.