Some older complex analysis textbooks state that $ \displaystyle \int_{0}^{\infty}e^{-x^{2}} \ dx$ can't be evaluated using contour integration.
But that's now known not to be true, which makes me wonder if you can ever definitively state that a particular real definite integral can't be evaluated using contour integration.
Edit: (t.b.) a famous instance of the above claim is in Watson, Complex Integration and Cauchy's theorem (1914), page 79:
Best Answer
There are such functions. For example, anything with infinitely many discontinuities. Take the Dirichlet function as an example; it is Lebesgue integrable, but one could not integrate it using the method of residues, which requires that there are only finitely many poles of the function on the real line.