Answer
$\dfrac{\ln (5/2)}{2}$
Work Step by Step
We need to set up the integration in the iterated form.
$I=\int_1^2 \int_{x}^{2x} \dfrac{y}{x^2+y^2}\ dy \ dx$
Let us suppose that $a=x^2+y^2 \implies da= 2ydy$
Next, we will solve the integral as follows:
$I=\int_1^2 [\int_{2x^2}^{5x^2} \dfrac{da}{2a}] \ dx$
or, $=\dfrac{1}{2} \int_1^2 [\ln (a)]_{2x^2}^{5x^2} \ dx$
or, $=\dfrac{1}{2} \times \ln (5/2) \times (2-1)$
or, $=\dfrac{\ln (5/2)}{2}$
