Help on solving integration

If only J and tau depend on the variable x we can assume that the left hand side is constant as well.

or

I notice that

1) you have two functions to solve for, but only one equation !!??

2) If x=0 the integral is zero. What is J(0) and tau(0) supposed to be?

As I assume that the constant on the right hand side should not be not zero, this means either J(0)->infinity and/or tau(0)->0

So I would say that one equation is not enough information and once you fix that, you may search the forum for "integral equation"
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and see if you can find something useful, as Prime has no facility out of the box to solve an integral equation.

You could differentiate your equation

with respect to x and with a little help from Prime's symbolics

you can write

You now need a second equation and then you could try to solve that system in a solve block with "odesolve".

I have no idea if this would work OK but it may be worth a try...