= Solution
For the <survivor functions>, $F_A'(u)=-f_A(u)$ and $F_B'(u)=-f_B(u)$. The <product rule> gives
$$
\frac{d}{du}[F_A(u)F_B(u)]=-f_A(u)F_B(u)-F_A(u)f_B(u).
$$
Integrating from 0 to $t$, using $F_A(0)=F_B(0)=1$, yields
$$
\boxed{\int_0^t f_A(u)F_B(u)\,du+\int_0^t F_A(u)f_B(u)\,du+F_A(t)F_B(t)=1.}
$$
The three terms partition the possible histories: A occurs first by time $t$, B occurs first by time $t$, or neither has occurred by $t$. The first two are the <cumulative incidence functions> for the two <competing risks>, while the third is the <survival probability> of their minimum. The initial value assumes nonnegative event times with no atom at zero.
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