List of definite integrals

In mathematics, the definite integral

is the area of the region in the xy-plane bounded by the graph of f, the x-axis, and the lines x = a and x = b, such that area above the x-axis adds to the total, and that below the x-axis subtracts from the total.

The fundamental theorem of calculus establishes the relationship between indefinite and definite integrals and introduces a technique for evaluating definite integrals.

If the interval is infinite the definite integral is called an improper integral and defined by using appropriate limiting procedures. for example:

A constant, such pi, that may be defined by the integral of an algebraic function over an algebraic domain is known as a period.

The following is a list of some of the most common or interesting definite integrals. For a list of indefinite integrals see List of indefinite integrals.

Definite integrals involving rational or irrational expressions

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Definite integrals involving trigonometric functions

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  (see Dirichlet integral)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Definite integrals involving exponential functions

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  (see also Gamma function)
 
 
 
 
 
  (the Gaussian integral)
 
 
 
 
 
  (where !! is the double factorial)
 
 
 
 
 
 
 
 
 
  (where   is Euler–Mascheroni constant)
 
 
 
 
 
 
 

Definite integrals involving logarithmic functions

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Definite integrals involving hyperbolic functions

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  holds if the integral exists and   is continuous.

See also

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References

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  • Reynolds, Robert; Stauffer, Allan (2020). "Derivation of Logarithmic and Logarithmic Hyperbolic Tangent Integrals Expressed in Terms of Special Functions". Mathematics. 8 (687): 687. doi:10.3390/math8050687.
  • Reynolds, Robert; Stauffer, Allan (2019). "A Definite Integral Involving the Logarithmic Function in Terms of the Lerch Function". Mathematics. 7 (1148): 1148. doi:10.3390/math7121148.
  • Reynolds, Robert; Stauffer, Allan (2019). "Definite Integral of Arctangent and Polylogarithmic Functions Expressed as a Series". Mathematics. 7 (1099): 1099. doi:10.3390/math7111099.
  • Winckler, Anton (1861). "Eigenschaften Einiger Bestimmten Integrale". Hof, K.K., Ed.
  • Spiegel, Murray R.; Lipschutz, Seymour; Liu, John (2009). Mathematical handbook of formulas and tables (3rd ed.). McGraw-Hill. ISBN 978-0071548557.
  • Zwillinger, Daniel (2003). CRC standard mathematical tables and formulae (32nd ed.). CRC Press. ISBN 978-143983548-7.
  • Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.