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Improper integral
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==Types of integrals== There is more than one theory of [[integral|integration]]. From the point of view of calculus, the [[Riemann integral]] theory is usually assumed as the default theory. In using improper integrals, it can matter which integration theory is in play. * For the Riemann integral (or the [[Darboux integral]], which is equivalent to it), improper integration is necessary ''both'' for unbounded intervals (since one cannot divide the interval into finitely many subintervals of finite length) ''and'' for unbounded functions with finite integral (since, supposing it is unbounded above, then the upper integral will be infinite, but the lower integral will be finite). * The [[Lebesgue integral]] deals differently with unbounded domains and unbounded functions, so that often an integral which only exists as an improper Riemann integral will exist as a (proper) Lebesgue integral, such as <math display="inline">\int_1^\infty \frac{dx}{x^2}</math>. On the other hand, there are also integrals that have an improper Riemann integral but do not have a (proper) Lebesgue integral, such as <math display="inline">\int_0^\infty \frac{\sin x}{x}\,dx</math>. The Lebesgue theory does not see this as a deficiency: from the point of view of [[measure theory]], <math display="inline">\int_0^\infty \frac{\sin x}{x}\,dx = \infty - \infty</math> and cannot be defined satisfactorily. In some situations, however, it may be convenient to employ improper Lebesgue integrals as is the case, for instance, when defining the [[Cauchy principal value]]. The Lebesgue integral is more or less essential in the theoretical treatment of the [[Fourier transform]], with pervasive use of integrals over the whole real line. * For the [[Henstock–Kurzweil integral]], improper integration ''is not necessary'', and this is seen as a strength of the theory: it encompasses all Lebesgue integrable and improper Riemann integrable functions.
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