Question

General question: What happens if basic integration rules, substitution, and integration by parts method ALL fail. Now what?

Answer 1

Depends. Numerical methods...

Answer 2

then you give up , you could try reimann sums for definite integrals but some functions dont have antiderivatives ( well as elementary functions anyway ) for example e^(x^2))

Answer 3

What is it was an indefinite integral? And how could you tell if some function had no antiderivatives?