Use Newton and a quadrature routine. That is, you are trying to find a root of
f(x) = x - \int_x^\infty A(z) dz
The derivative of this is f'(x) = 1 + A(x). So, you can just repeatedly take Newton steps
x \leftarrow x - \frac{f(x)}{f'(x)}
where f(x) is evaluated with a quadrature routine like QuadGK:
using QuadGK
f(x) = x - quadgk(A, x, Inf, rtol=1e-4)[1]
for example. (QuadGK handles the semi-infinite integral for you by a coordinate transformation.)