Numerical evidence suggests:


$$ \int\_0^{\frac12}\int\_0^{\frac12}\frac{1}{1-x^2-y^2} dy \, dx= \frac{G}{3}\qquad (1)$$


Couldn't find the indefinite integral, though maple simplifies (1) to


$$ \int \_{0}^{1/2}\!-\arctan \left( 1/2\,{\frac {1}{\sqrt {-1+{x}^{2}}}} \right) {\frac {1}{\sqrt {-1+{x}^{2}}}}{dx}$$



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> Is (1) true?
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Conjectured integral for Catalan's constant
Mathematica confirms the following:


Change the integral to polar coordinates to get


$$\frac{(1)}{2} = \int\_0^{\pi/4} \int\_0^{\sec(\theta)/2} \frac{1}{1-r^2}r\ dr\ d\theta = \frac{G}{6}.$$


