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Is it possible to evaluate this integral in spacetime?

$$\int_{\Sigma} \frac{dydz}{[a_{o}(y^{2}+z^{2})+2f_{o}y+2g_{o}z+c_{o}]^{2}}$$ where $$a_{o}c_{o}-f_{o}^{2}-g_{0}^{2}=\frac{1}{4}.$$

If it is possible, can someone shed some light on how can this be done?

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  • $\begingroup$ If this is over spacetime, should the measure be $\sqrt{-g}d^4x$? $\endgroup$ – Gradient137 Nov 10 '18 at 7:02
  • $\begingroup$ Have you tried shifting $y,\,z$ by the obvious constants? $\endgroup$ – J.G. Nov 10 '18 at 7:03
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I'll offer a hint, expanding on my comment. Define $y':=y+\frac{f_0}{a_0},\,z':=z+\frac{g_0}{a_0}$. The quadratic becomes $$a_0(y'^2+z'^2)+c_0-\frac{f_0^2+g_0^2}{a_0}=a_0\bigg(y'^2+z'^2+\tfrac{1}{(2a_0)^2}\bigg).$$So now you just need to evaluate$$\frac{1}{a_0^2}\int_\Sigma\frac{dy'dz'}{\bigg(y'^2+z'^2+\tfrac{1}{(2a_0)^2}\bigg)^2}\cdot$$The substitution $u=y'^2+z'^2$ might be worthwhile.

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