In the book Numerical methods using MATLAB (John H. Mathews, Kurtis D. Fink) I found this exercise as part of Algorithms and Programs section:
First of all: what is the meaning of an aproximate solution in this case?
More important: how can I solve it?
and finally, is a general way to solve this kind of integral equation for, lets say $$v(x)=f(x)+\int_{-1}^{1}U(x,t)v(t)dt$$
Substitute the integral by the Legendre-quadrature with nodes $t_i$ and weights $w_i$, and consider the cases where $x$ equals each of the nodes:
$$\begin{align*} \forall x\in\{t_1,t_2,\ldots,t_n\}:\quad v(x)&=f(x)+\int_{-1}^{1}U(x,t)v(t)dt\\ &\approx f(x)+\sum_{i=1}^n w_iU(x,t_i)v(t_i) \end{align*}$$
That gives you $n$ linear equations:
$$\begin{align*}\forall j\in [1,n]:\quad v(t_j)=f(t_j)+\sum_{i=1}^n w_iU(t_j,t_i)v(t_i) \end{align*}$$
with $n$ unknowns $v(t_1),v(t_2),\ldots,v(t_n)$.
When they are solved, everything but $x$ is known in $\ v(x)\approx f(x)+\sum_{i=1}^n w_iU(x,t_i)v(t_i)\ $