Answer
$[1-\ln (2)] \pi $
Work Step by Step
Our aim is to integrate the integral as follows:
$\int ^0_{-1} \int ^0_{-\sqrt{1-x^2}}\dfrac{2}{1+\sqrt{x^2+y^2}} \space dy \space dx $
or, $=\int^{3\pi/2}_{\pi} \int^1_0\dfrac{2r}{1+r} \space dr \space d\theta $
or, $=2\int^{3\pi/2}_{\pi} \int^{1}_0(1-\dfrac{1}{1+r}) \space dr \space d\theta $
or, $=2\int ^{3\pi/2}_\pi (1-\ln (2)) \space d\theta $
or, $=[1-\ln (2)] \pi $
