Question

The value of the definite integral ${\int }_{\frac{1}{b}}^b\frac{xlnx}{(a^2+x^2)(a^2{x}^2+1)}dx$ (where b > 0) equals

Answer 1

Put $x=\frac{1}{t}\Rightarrow dx=-\frac{1}{t^2}dt$
$I={\int }_b^{\frac{1}{b}}\frac{tlnt}{(a^2{t}^2+1)(a^2+t^2)}dt=-{\int }_{\frac{1}{b}}^b\frac{tlntdt}{(a^2{t}^2+1)(a^2+t^2)}I=-I\Rightarrow I=0$