Error with ForwardDiff 1 and QuadGK

Hi ForwardDiff team,

I observe an error when updating from ForwardDiff v0.10 to v1.x. The error occurs when differentiating through quadgk integration.

A MWE to reproduce the error is as follows:

using ForwardDiff
using QuadGK

delta = [ 1.0, 10.0 ]
chi = [ 0.03, 0.30 ]
S = inv([ exp(-chi_ * delta_) for delta_ in delta, chi_ in chi ])

g(x) = quadgk(t -> S*x, 0.0, 1.0)[1]

ForwardDiff.jacobian(g, zeros(2))

With ForwardDiff 1.3, I get the following error:

ERROR: LoadError: DomainError with 0.5:
integrand produced Dual{ForwardDiff.Tag{typeof(g), Float64}}(NaN,NaN,NaN) in the interval (0.0, 1.0)

Note that the code does not throw with ForwardDiff v0.10.39.

Also note that when changing the matrx S, e.g. adding a line S .+= 0.5 then above code also works with v1.3.

Would be great if you could have a look.

Start by running the provided MWE with ForwardDiff 1.3 and QuadGK, then compare it with ForwardDiff v0.10.39. Investigate the differentiation path through quadgk and the effect of the matrix S values; done means the example no longer produces Dual NaN values or a DomainError.