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Is there a method to check whether a function is integrable?

Of-course trying to solve it is one but some questions in integration may be so tricky that I don't get the correct method to start off with those problems. So, is there a method to find correctly whether a function is integrable?

Clarification: I am asking about indefinite integrals which have no elementary anti derivative.

Rajath Radhakrishnan
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    By integrable, do you mean "has a convergent definite integral", or "has an elementary anti-derivative"? For example the integral of $\operatorname{e}^{-x^2}$ over $\mathbb{R}$ is $\sqrt{\pi}$, while there is no elementary function whose derivative is $\operatorname{e}^{-x^2}$. – Fly by Night Sep 13 '13 at 15:21
  • I am asking about indefinite integrals which has an elementary anti derivative. – Rajath Radhakrishnan Sep 13 '13 at 15:26

2 Answers2

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Maybe you are asking for a procedure for determining, given an elementary function $f(x)$, whether there is an elementary function $F(x)$ such that $F'(x)=f(x)$.

With some caveats, there is such a procedure, called the Risch Algorithm. For some discussion of the algorithm, please see the linked Wikipedia article.

Part-implementations of the Risch Algorithm are a component of various symbolic integration programs.

André Nicolas
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If you mean Riemann integrable on an interval $[a,b]$, you may find Lebesgue's criterion useful: $f$ is Riemann integrable on $[a,b]$ if and only if it is continuous almost everywhere (i.e. its discontinuities form a set of measure $0$) and bounded.

Robert Israel
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