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Though there exists an infinite number of functions and area under them in a given interval is finite, it is impossible to determine the area using integration. Such functions are called "non-integrable functions". How to identify the given function as non-integrable?

For example : It is impossible to find $\int_0^1 x^x \, dx$

Srivatsav
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  • FYI, "non-integrable" is not a good term to use. That said, experience with standard integration techniques allows educated guesses to be made, but a proof would be needed. However, if the integral is simple enough, then one can sometimes prove (by using a substitution or other method) that explicit integrability of the integral under consideration implies explicit integrability of an integral known not to be explicitly integrable (analogous to proving $3\sqrt 2$ is irrational when you're allowed to use the fact that $\sqrt 2$ is irrational). – Dave L. Renfro May 02 '22 at 16:42
  • If you're OK with series, then $\int_0^1 x^x dx = -\sum_{n=1}^\infty (-n)^{-n}$ is an explicit expression for it. – eyeballfrog May 02 '22 at 19:41

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