Question

Evaluate the definite integral ${\int }_2^3\frac{xdx}{x^2+1}$

Answer 1

Let $I={\int }_2^3\frac{xdx}{x^2+1}$
$\int \frac{x}{x^2+1}dx=\frac{1}{2}\int \frac{2x}{x^2+1}dx=\frac{1}{2}\log (1+x^2)=F\left(x\right)$
By second fundamental theorem of calculus, we obtain
$I=F\left(3\right)-F\left(2\right)$
$=\frac{1}{2}\left[\log \right(1+\left(3{)}^2\right)-\log (1+(2{)}^2\left)\right]$
$=\frac{1}{2}\left[\log \right(10)-\log (5\left)\right]$
$=\frac{1}{2}\log \left(\frac{10}{5}\right)=\frac{1}{2}\log 2$