Transformer Integrals

$\ln(xy)=(\ln{x})(\ln{y})$

Converting to polar form

$\implies r=\dfrac{\exp\left(1-\dfrac{1}{2}\sqrt{\ln^2\tan{\theta}+4}\right)}{\sqrt{\sin{\theta}\cos{\theta}}}$

Required area $=A=\dfrac{1}{2}\displaystyle\int_0^{\frac{\pi}{2}}r^2\,d\theta$

$=\displaystyle\int_0^{\frac{\pi}{4}}\csc{\theta}\sec{\theta}\exp\left(2-\sqrt{\ln^2\tan{\theta}+4}\right)\,d\theta$

Let $\ln\tan{\theta}=t\,\implies\,dt=\csc{\theta}\sec{\theta}\,d\theta$

$\implies A=\displaystyle\int_{-\infty}^0 e^{2-\sqrt{t^2+4}}\,dt=\displaystyle\int_0^\infty e^{2-\sqrt{t^2+4}}\,dt$

By Inverting the function, the integral becomes:

$\displaystyle\int_0^1\sqrt{\ln^2x-4\ln{x}}\,dx$

$\implies \boxed{\alpha=4}$