Find the root of integral equation

[TheSource007]

Homework Statement

Hi everyone.
I have encountered a weird equation while doing some research and I have no idea how to solve it.
The equation goes like this

∫ dR / (1+ c*r) ^ (a/r) = d, limits of integration are from 0 to Rmax,
where Rmax ^2 = ^2 - α^2, where u is a constant value of r.
and r^2 = R^2 + α^2,
c is a constant and d is my independent variable.

Homework Equations

The Attempt at a Solution

I want to do a C function that takes one value of d and returns a value of α. I thought of doing
∫ (1 / (1+ c*r) ^ (a/r) ) - d = 0 and try to find the root, or do minimization. The problem is that α also appears on the limits of integration, and even if it wasn't there, I still don't know how to do either.
Also, for any value of d, there is a value of α, but r changes from r = α to r= u.

I would appreciate any help. Thanks

 

[haruspex]

Even though alpha appears both inside the integral and in the limit, you can differentiate wrt alpha. That allows you to use standard iterative methods: given a trial alpha, find the resulting d, calculate the error and use the derivative to estimate an adjustment to alpha.
... and it will have the standard pitfalls.

 

[TheSource007]

Could you refer me to sources where they explain more on the iterative methods that can do this sort of problem? I have no experience in finding a numerical value for an unknown that appears inside an integral (and inside the limits for that matter)

 

[haruspex]

Could you refer me to sources where they explain more on the iterative methods that can do this sort of problem? I have no experience in finding a numerical value for an unknown that appears inside an integral (and inside the limits for that matter)

A basic method when you know the derivative function is Newton-Raphson. See e.g. http://en.wikipedia.org/wiki/Newton's_method.
You understand how to expand $\frac d{dx}\int^{f(x)}g(x, z).dz$, right?

 

[TheSource007]

You understand how to expand $\frac d{dx}\int^{f(x)}g(x, z).dz$, right?

Is that just the fundamental theorem of calculus?

And also, is there any method to find the root that does not involve a first trial of alpha?
Thanks

 

[haruspex]

Is that just the fundamental theorem of calculus?

It's a little more complicated because x is in two places.

And also, is there any method to find the root that does not involve a first trial of alpha?
Thanks

No.