Need Help Understanding How To Integrate With An Implicit Variable My calculus is really rusty (dang Mathematica/Matlab), and I was wondering if anyone could help me with an equation I am having trouble integrating. I have attached a snapshot of the paper that I am trying my best to understand below:enter image description here


Now, I am able to following everything, but I am confused when it comes to equation 22. I am not exactly sure how to integrate this; what to do with psi in the denominator of the integral. Since the integration is with respect to ds' I know the psi would not matter and could be taken out of the integral, but that would make things too easy and I know its wrong. Any ideas/help??
 A: First things first:

that would make things too easy

That's not a reason to be wrong, so be wary of saying something is too easy to be correct.
However, in this case, there is more to it than that. $\psi$ can be taken out of the integral if it doesn't depend on the integration variable $s'$, but in this case it does depend on $s'$ in the manner given by equation 18.
$$s - s' = \frac{R\psi^3}{24}\frac{\psi + 4x}{\psi + x}$$
In general, the easiest way to handle something like this is to see if you can find an explicit expression for $\psi$ as a function of the integration variable $s'$ and plug it in. You'd want to do the same for $\lambda$, if you can. That way, you get
$$\int \frac{1}{\text{function of }s' + 2x}(\text{other function of }s')ds'$$
so the expression being integrated is in terms of $s'$ and things that don't depend on $s'$. Then you can go ahead and try to do the integral in the normal way.
If that doesn't work out, you have to use some sort of "trick" (i.e. something other than directly integrating), but there is a very long list of those and you'd have to figure out which applies on a case-by-case basis. Perhaps you can take some motivation from the result they get in the paper.
