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$\frac{1}{c^2}\int_{\theta=-\pi/2}^{\theta=\pi/2} \frac{kM}{r^2}cos\theta ds=2\frac{kM}{c^2\Delta}$

An observation: $\cos\theta=\frac{\Delta}{r}$

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The entire calculation is done in an early section of the answere here: – Ron Maimon May 31 '12 at 19:42
Thanks! I've read quite a bit of the other sections as well. – MadScientist Jun 1 '12 at 2:01
up vote 1 down vote accepted


$s = \Delta \tan\theta \quad \Rightarrow \quad ds = \frac{\Delta}{\cos^2\theta}d\theta $


$r = \frac{\Delta }{\cos \theta }$

and insert it into your integral. What you get is

$\frac{1}{c^2} \intop_{-\pi/2}^{\pi/2}\frac{kM}{\Delta } \cos\theta d\theta = 2 \frac{kM}{c^2 \Delta} $.

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