What are examples of functions that cannot be integrated?
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
Other functions, such as a function
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.
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