Solution
= Solution
The event contribution is an ordinary density integral. Since $f=-F'$ and $H'=h$, <integration by parts> gives
$$
\begin{aligned}
\mathbb E[H(X)\mathbf1_{\{T\le c\}}]
&=\int_0^c H(t)f(t)dt=-\int_0^cH(t)\,dF(t)\\
&=-H(c)F(c)+H(0)F(0)+\int_0^cF(t)h(t)dt\\
&=-H(c)F(c)+\int_0^cf(t)dt.
\end{aligned}
$$
Consequently
$$
\boxed{\mathbb E[H(X)\mathbf1_{\{T\le c\}}]=1-F(c)-F(c)H(c).}
$$