Question:
$\int_{2}^{3} \frac{x d x}{x^{2}+1}$
Solution:
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
$I=\mathrm{F}(3)-\mathrm{F}(2)$
$=\frac{1}{2}\left[\log \left(1+(3)^{2}\right)-\log \left(1+(2)^{2}\right)\right]$
$=\frac{1}{2}[\log (10)-\log (5)]$
$=\frac{1}{2} \log \left(\frac{10}{5}\right)=\frac{1}{2} \log 2$