Get numerical solution of integral equation with parameters?

Questions
1

Hello! Could you tell me about how to take the next numerical calculation in mathematica? (perhaps there are special packages).
I have an expression (in reality slightly more complex):

V=x^2 + \int_a^b l \sqrt{l^2-m^2} \left(\text Log \left(e^{-\left(\beta \left(\sqrt{\left(\sqrt{l^2-m^2}+U\right)^2+(m+x)^2+N}+u\right)\right)}+1\right)\right) \, dl 

 V = x^2 + 
   Integrate[
     l*Sqrt[l^2 - m^2]*
     Log*(1 + 
       e^(-(\[Beta]*(u + 
         Sqrt[(Sqrt[l^2 - m^2] + U)^2 + (m + x)^2 + N])))), {l, a, b}]

where x is function of l; m, N are constants; \beta, u, U are parameters.
I need to find the dependence U on u and \beta (in order to draw graph) from an equation:
\frac {\partial V} {\partial x}=0
(x will be needed to set a constant after differentiation; In reality,there is not the derivative, but a variation)

If I have an integral (without parameters) rather than equation, I would try to do the following ones:

  1. to define the a region of integration (due to graphical representation of function)
  2. to tabulate integrand
  3. to calculate the integral that is to get a number.

Nevertheless I have the equation, which probably requires other method. I would appreciate Mathematica literature on this subject, or help.

I do not know how actual it is to calculate integral equation, but the integral can be led to another kind:
l \sqrt{l^2-m^2} \left(\text {Log} \left(e^{-\left(\beta \left(\sqrt{\left(\sqrt{l^2-m^2}+U\right)^2+(m+x)^2+N}+u\right)\right)}+1\right)\right) \to
\left(l^2-m^2\right)^{3/2} \frac{\text{\cosh(\beta u)} +\exp \left(-\beta \sqrt{\left(\sqrt{l^2-m^2}+U\right)^2+(m+x+y)^2+(q+z)^2}\right)}{\text{\cosh (\beta u)} -\cosh \left(\beta \sqrt{\left(\sqrt{l^2-m^2}+U\right)^2+(m+x+y)^2+(q+z)^2}\right)} 

(x^2-m^2)^(3/2) (cosh(\[Beta]u)+exp(-\[Beta] Sqrt[(Sqrt[l^2-m^2]+U)^2+(m+x+y)^2+(z+q)^2]))/(cosh(\[Beta]u)-cosh(\[Beta] Sqrt[(Sqrt[l^2-m^2]+U)^2+(m+x+y)^2+(z+q)^2]))

Perhaps it was necessary to start with something simpler. Let

V=x^2+\int_{a}^{b} x u U \beta l dl

\frac {\partial V} {\partial x}=0

x is assumed a constant after differentiating.

\int_a^b \beta l u U \, dl+2 x=0

U=\frac{4 x}{\beta u \left(a^2+b^2\right)}

In all I get the dependency U, on u and \beta

I need to do the