What Defines an Improper Integral?

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SUMMARY

Improper integrals are defined by the violation of specific conditions during evaluation. An integral of the form ##\int\limits^b_a f(x)dx## is considered improper if either the function f has an infinite discontinuity within the interval ##[a, b]## or if the interval itself is infinite. The concept of convergence is crucial in determining the validity of an improper integral. Understanding these definitions is essential for correctly evaluating integrals that do not conform to standard bounded conditions.

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  • Understanding of integral calculus
  • Familiarity with limits and convergence
  • Knowledge of continuous and discontinuous functions
  • Basic proficiency in evaluating definite integrals
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NihalRi
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I find it easy to picture bounder integrals because they are the area under the graph but when it is unbounded what is it exactly. If we evaluate it we find the equation of a many possible graphs where the derivative is the function that was originally in the integral. How did we get from that to area?
 
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Hey NihalRi.

For improper integrals we look at limits and require that the integral have some sort of convergence.
 
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While evaluating ##\int\limits^b_a f(x)dx## there was an assumption that
(1) f does not have an infinite discontinuity in ##[a, b]##
(2) The integral interval ##[a,b]## is not infinite i.e, both a and b are finite values. (Edited by mentor)
If either of the above 2 condition is violated we say that the integral ##\int\limits^b_a f(x)dx## is an improper integral.
 
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