Integration methods...an error-proof method?

[AlexFekken]

Let me rephrase my earlier response: if your aim is to get an accurate approximation of the integral then you should not (use algorthims that) avoid "singularities", but you need to embrace, understand and deal with them.

Simply avoiding them (i.e. sweeping them under the rug) will give you an answer, but no indication as to how (in)accurate that answer is.