The paper presents an adaptive method for the evaluation of multidimensional integrals over the unit cube. The measure used to partition the domain is suited for integrands which are monotonic in each dimension individually, and is therefore suitable for problems stemming from finance where this is often the case. We use a QMC method for each sub-problem resulting from the partitioning of the domain. The article is part one of a work on this topic, and presents the method together with various local variance reduction techniques. The material is presented with an alignment to option pricing problems. In the companion paper we present an option pricing problem and simulation results on different setups of this. We compare the convergence properties of the adaptive method with the convergence properties of the QMC method used directly on the problem. We find that the adaptive method in many configurations outperform the conventional QMC method, and we develop criteria on the problem for when the adaptive method can be expected to outperform the conventional.