Prove

Let $I=\int_{2}^{3} \frac{x}{x^{2}+1} d x$

$\int \frac{x}{x^{2}+1} d x=\frac{1}{2} \int \frac{2 x}{x^{2}+1} d x=\frac{1}{2} \log \left(1+x^{2}\right)=\mathrm{F}(x)$

By second fundamental theorem of calculus, we obtain

$I=\mathrm{F}(3)-\mathrm{F}(2)$

$=\frac{1}{2}\left[\log \left(1+(3)^{2}\right)-\log \left(1+(2)^{2}\right)\right]$

$=\frac{1}{2}[\log (10)-\log (5)]$

$=\frac{1}{2} \log \left(\frac{10}{5}\right)=\frac{1}{2} \log 2$