Answer
$[\ln (4) -1] \pi $
Work Step by Step
$I= \int_{-1}^{1} \int_{-\sqrt {1-y^2}}^{\sqrt {1-y^2}} \ln (x^2+y^2+1) \ dx \ dy$
or, $= \int_{0}^{2 \pi} \int_{0}^{1} r \ln (r^2+1) \ dr \ d \theta$
Consider $r^2+1 = u \implies r dr =du$
So, $I=\dfrac{1}{2} \int_{0}^{2 \pi} \int_{1}^2 (\ln u du) d \theta \\=\dfrac{1}{2} \int_{0}^{2 \pi} [u\ln (u) - u]_1^2 \ d\theta \\=\int_{0}^{2 \pi} \dfrac{(2 \ln 2 -1) )}{2} \ d\theta \\=[\ln (4) -1] \pi $
