I am trying to integrate

where is a constant. I am trying to obtain an expression that can be plotted on a versus p chart.

Attempt at a solution:

My naive attempt was to simply integrate wrt p and I obtained

which then solves to

Unfortunately when plotted it seems unphysical, and I think I may have done the integration wrong. I suspect the problem is that the two variables p and are not independent and multivariable integration is required, but I have no experience of that and no idea where to start. Can anyone help?

Background:

References http://www.phys.au.dk/~fedorov/GTR/09/note11.pdf and Lemaitre metric - Wikipedia, the free encyclopedia

I am using the simplified version of the Lemaitre metric

which ignores angular motion and I am trying to obtain an expression for light paths which always have ds=0, so that for light:

and from the references

which I substituted into the dp/d equation to obtain the expression that I am trying to integrate. If it helps any, the value 2 can be substituted for . Any help would be greatly appreciated.