List of definite integrals

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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[edit]

Definite integrals involving trigonometric functions[edit]

(see Dirichlet integral)

Definite integrals involving exponential functions[edit]

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

Definite integrals involving logarithmic functions[edit]

Definite integrals involving hyperbolic functions[edit]

Frullani integrals[edit]

holds if the integral exists and is continuous.

See also[edit]


  • 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.