Question

Evaluate: ${\int }_2^3\frac{xdx}{x^2+1}$.

Answer 1

${\int }_2^2\frac{xdx}{x^2+1}={\int }_{5}10\frac{\frac{dt}{2}}{t}$ Put $x^2+1=t$
$=\frac{1}{2}{\int }_5^5\frac{dt}{t}=\frac{1}{2}\log |t|{|}_5^{10}2xdx=dt\Rightarrow xdx=\frac{dt}{2}$
$=\frac{1}{2}[\log 10-\log 5]$ when $x=2\Rightarrow t=5$
when $x=3\Rightarrow t=10$
$=\frac{1}{2}\left(\log \frac{10}{5}\right)=\frac{1}{2}\log 2$.