Home > Calculus > Formal Definition of the Definite Integral

What is an improper integral?

Answer 1

The definite integral over interval #[a,b]# of #f# is initially defined For a function #f# that includes #[a,b]# in its domain.

That is: we start with a function #f# that is defined for all #x in [a,b]#

Improper integrals extend the initial definition by allowing #a#, or #b#, or both to be outside the domain of #f# (but on the 'edge' so we can look for limits) or for the interval to lack left and/or right endpoints (infinite intervals).

Examples:

#int_0^1 lnx dx# #color(white)"sssssssssss"# integrand not defined at #0#

#int_5^7 1/(x^2-25) dx# #color(white)"ssssss"# integrand not defined at #5#

#int_1^oo 1/x^2 dx# #color(white)"sssssssssss"# interval has no right endpoint

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Answer 2

Axel Alvarez

An improper integral is an integral with one or both limits of integration being infinite or where the function being integrated has a discontinuity within the interval of integration. These integrals cannot be evaluated using the standard methods and require special techniques such as limit processes to determine their convergence or divergence.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.