We consider the maximization version of the edge-disjoint path problem (EDP). In undirected graphs and directed acyclic graphs, we obtain an $O(\sqrt{n})$ upper bound on the approximation ratio where $n$ is the number of nodes in the graph. We show this by establishing the upper bound on the integrality gap of the natural relaxation based on multicommodity flows. Our upper bound matches within a constant factor a lower bound of $\Omega(\sqrt{n})$ that is known for both undirected and directed acyclic graphs. The best previous upper bounds on the integrality gaps were $O(\min\{n^{2/3}, \sqrt{m}\})$ for undirected graphs and $O(\min\{\sqrt{n \log n}, \sqrt{m}\})$ for directed acyclic graphs; here $m$ is the number of edges in the graph. These bounds are also the best known approximation ratios for these problems. Our bound also extends to the unsplittable flow problem (UFP) when the maximum demand is at most the minimum capacity.