Note also that if your decay term is slow (relative to the oscillation period), then direct methods (not specialized for the oscillation term) correspond to very ill-conditioned sums that are subject to large cancellation errors — basically, you have a lot of positive and negative contributions that are supposed to exactly cancel, but don’t because of roundoff and other errors.

Even with exponential decay, if the rate is slow enough then you could run into trouble with non-specialized integration schemes.

A classic example of this is asymptotics of Fourier transforms — even if you have a nice smooth, rapidly decaying function f(x), if you want to compute high-frequency asymptotics of its Fourier transform F(k) (which goes to zero for large k, but maybe you want to know just how small it is), then you often need analytical tricks such as saddle-point methods, and/or specialized quadrature schemes.