Conditions for integrability Michael Spivak, in his "Calculus" writes 

Although it is possible to say precisely which functions are integrable,the criterion for integrability is too difficult to be stated here

I request someone to please state that condition.Thank you very much!

This is commonly called the Riemann-Lebesgue Theorem, or the Lebesgue Criterion for Riemann Integration (the wiki article).
The statement is that a function on $[a,b]$ is Riemann integrable iff


*

*It is bounded

*It is continuous almost everywhere, or equivalently that the set of discontinuities is of zero lebesgue measure

