  Let $g(z)=\int_0^zp(t)\exp(q(t))\,dt+c$ where $p,q$ are polynomials and
$c\in\mathbb{C}$, and let $f$ be the function from Newton's method for $g$. We
show that under suitable assumptions the Julia set of $f$ has Lebesgue measure
zero. Together with a theorem by Bergweiler, our result implies that $f^n(z)$
converges to zeros of $g$ almost everywhere in $\mathbb{C}$ if this is the case
for each zero of $g''$. In order to prove our result, we establish general
conditions ensuring that Julia sets have Lebesgue measure zero.
