We study the logarithmic error of numerical methods for the integration of monotone or unimodal positive functions. We compare adaptive and nonadaptive methods in the worst case setting. It turns out that adaption significantly helps for the class of unimodal functions, but it does not help for the class of monotone functions. We do not assume any smoothness properties of the integrands and obtain numerical methods that are reliable even for discontinuous integrands. Numerical examples show that our method is very competitive in the case of nonsmooth and/ or peak functions.