Solution
= Solution
With cause-specific hazards $h_A$ and $h_B$, survival free of either event through time $u$ is
$$
S(u)=\exp\!\left[-\int_0^u\{h_A(s)+h_B(s)\}\,ds\right].
$$
The probability of remaining event-free to $u$ and then experiencing $A$ in $du$ is $S(u)h_A(u)du$. Therefore
$$
F_A(t)=\int_0^t
\exp\!\left[-\int_0^u\{h_A(s)+h_B(s)\}\,ds\right]
h_A(u)\,du.
$$