  In this paper, we establish a local Lie theory for relative Rota-Baxter
operators of weight $1$. First we recall the category of relative Rota-Baxter
operators of weight $1$ on Lie algebras and construct a cohomology theory for
them. We use the second cohomology group to study infinitesimal deformations of
relative Rota-Baxter operators and modified $r$-matrices. Then we introduce a
cohomology theory of relative Rota-Baxter operators on a Lie group. We
construct the differentiation functor from the category of relative Rota-Baxter
operators on Lie groups to that on Lie algebras, and extend it to the
cohomology level by proving the Van Est theorem between the two cohomology
theories. We integrate a relative Rota-Baxter operator of weight 1 on a Lie
algebra to a local relative Rota-Baxter operator on the corresponding Lie
group, and show that the local integration and differentiation are adjoint to
each other. Finally, we give two applications of our integration of Rota-Baxter
operators: one is to give an explicit formula for the factorization problem,
and the other is to provide an integration for matched pairs.
