Question

What are examples of functions that cannot be integrated?

Answer 1

Answer 2

It depends on what you mean. Do you mean you can't find a formula for an antiderivative? Or do you mean the definite integral doesn't exist?

Some functions, such as `sin(x^2)`, have antiderivatives that don't have simple formulas involving a finite number of functions you are used to from precalculus (they do have antiderivatives, just no simple formulas for them). Their antiderivatives are not "elementary".

Other functions, such as a function `f(x)` that equals 1 when `x` is rational and 0 when `x` is irrational are not "Riemann integrable" over any closed interval `[a,b]`. The problem lies in the fact that, for a given partition of the interval, you can always pick sample points that are either all irrational or all rational, which will lead to sums that don't converge to the same answer as the subintervals all get smaller.

This last function is, however, "Lebesgue integrable" (pronounced "Lah-bagh" with a long "a" sound in the second syllable). I won't get into details, but in a nutshell, there are lots of "theories of integration" with respect to which a given function might be integrable or not.