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 the most common definite integrals. For a list of indefinite integrals see List of indefinite integrals.

Definite integrals involving rational or irrational expressions

Definite integrals involving trigonometric functions

(see Dirichlet integral)

Definite integrals involving exponential functions

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

Definite integrals involving logarithmic functions

Definite integrals involving hyperbolic functions

Frullani integrals

holds if the integral exists and is continuous.

See also


  • 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. 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.