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In calculus, an improper integral is the limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number or or or, in some cases, as both endpoints approach limits. Such an integral is often written symbolically just like a standard definite integral, perhaps with ''infinity'' as a limit of integration. Specifically, an improper integral is a limit of the form : or of the form : in which one takes a limit in one or the other (or sometimes both) endpoints . When a function is undefined at finitely many interior points of an interval, the improper integral over the interval is defined as the sum of the improper integrals over the intervals between these points. By abuse of notation, improper integrals are often written symbolically just like standard definite integrals, perhaps with ''infinity'' among the limits of integration. When the definite integral exists (in the sense of either the Riemann integral or the more advanced Lebesgue integral), this ambiguity is resolved as both the proper and improper integral will coincide in value. Often one is able to compute values for improper integrals, even when the function is not integrable in the conventional sense (as a Riemann integral, for instance) because of a singularity in the function, or poor behavior at infinity. Such integrals are often termed "properly improper", as they cannot be computed as a proper integral. ==Examples== The original definition of the Riemann integral does not apply to a function such as on the interval } &^ \frac \int_^ \frac = \lim_ 3(1-\sqrt()) + \lim_ 3(1-\sqrt()) \\ & But the similar integral : cannot be assigned a value in this way, as the integrals above and below zero do not independently converge. (However, see Cauchy principal value.) 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Improper integral」の詳細全文を読む スポンサード リンク
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