The problem I am trying to solve is $\max_{k \in \mathbb{N}} \int_0^1 u d F(u)^k - ck$, where the associated utility is an iid random variable U following $F(\cdot)$ on [0,1]. $c > 0$ is the constant marginal cost of search. By choosing $k \in \mathbb{N}$, the decision maker knows the realized $u_1, u_2, \ldots , u_k$ and chooses the highest alternative.
The simplification I am having trouble with is at the end of this line of logic:
"Notice that for each k, the benefit of search is given by: $\int_0^1 u d F(u)^k = 1 - \int_0^1 F(u)^k$ du$"
This seems to me to have to be integration by parts I'm just not sure where the 1 is coming from.
$$ \int_0^1 d(uF^k(u)) = \left[uF^k(u)\right]_0^1 = \int_0^1 u dF^k(u) + \int_0^1 F(u)^k du $$ this is leads to $$ \left[uF^k(u)\right]_0^1 = 1\cdot F^k(1) - 0\cdot F^k(0) = 1 = \int_0^1 u dF^k(u) + \int_0^1 F(u)^k du $$ re-arrange $$ \int_0^1 u dF^k(u) = 1 - \int_0^1 F(u)^k du $$