above, which is particularly useful when the algebraic manipulations involved are very complex or lengthy. Integrals which have already been derived can be looked up in a table of integrals. Of non-continuous functions Non-continuous functions can have antiderivatives. While there are still open questions in this area, it is known that: Some highly pathological functions with large sets of discontinuities may nevertheless have antiderivatives. In some cases, the antiderivatives of such pathological functions may be found by Riemann integration, while in other cases these functions are not Riemann integrable. Assuming that the domains of the functions are open intervals: A necessary, but not sufficient, condition for a function to have an antiderivative is that have the intermediate value property. That is, if is a subinterval of the domain of and is any real number between and , then there exists a between and such that . This is a consequence of Darboux's theorem. The set of discontinuities of must be a meagre set. This set must also be an F-sigma set (since the set of discontinuities of any function must be of this type). Moreover, for any meagre F-sigma set, one can construct some function having an antiderivative, which has the given set as its set of discontinuities. If has an antiderivative, is bounded on closed finite subintervals of the domain and has a set of discontinuities of Lebesgue measure 0, then an antiderivative may be found by integration in the sense of Lebesgue. In fact, using more powerful integrals like the Henstock–Kurzweil integral, every function for which an antiderivative exists is integrable, and its general integral coincides with its antiderivative. If has an antiderivative on a closed interval , then for any choice of partition if one chooses sample points as specified by the mean value theorem, then the corresponding Riemann sum telescopes to the value . However if is unbounded, or if is bounded but the set of discontinuities of has positive Lebesgue measure, a different choice of sample points may give a significantly different value for the Riemann sum, no matter how fine the partition. See Example 4 below. Some examples Basic formulae If , then . See also Antiderivative (complex analysis) Formal antiderivative Jackson integral Lists of integrals Symbolic integration Area Notes References Further reading Introduction to Classical Real Analysis, by Karl R. Stromberg; Wadsworth, 1981 (see also) Historical Essay On Continuity Of Derivatives by Dave