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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.
Integration is the basic operation in integral calculus.While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful.
Using Euler's formula, any trigonometric function may be written in terms of complex exponential functions, namely and and then integrated. This technique is often simpler and faster than using trigonometric identities or integration by parts , and is sufficiently powerful to integrate any rational expression involving trigonometric functions.
Abramowitz, Milton; Stegun, Irene A., eds. (1972). "Chapter 3". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.
The integration formula for double integrals may be generalized to any multiple integral. In all cases, there is a parameter value n ∗ {\textstyle n^{\ast }} or array of parameter values N ∗ {\textstyle N^{\ast }} that solves one or more linear equations derived from the exponent terms of the integrand's series expansion.
In mathematics, there are two types of Euler integral: [1]. The Euler integral of the first kind is the beta function (,) = = () (+); The Euler integral of the second kind is the gamma function [2] =
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Because this is undefined when x = −b / a, the most general form of the antiderivative replaces the constant of integration with a locally constant function. [1] However, it is conventional to omit this from the notation.