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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.
Integration, the process of computing an integral, is one of the two fundamental operations of calculus, [a] the other being differentiation. Integration was initially used to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Usage of integration expanded to a wide ...
For example, one method of solving a boundary value problem is by converting the differential equation with its boundary conditions into an integral equation and solving the integral equation. [1] In addition, because one can convert between the two, differential equations in physics such as Maxwell's equations often have an analog integral and ...
Integration by parts is often used in harmonic analysis, particularly Fourier analysis, to show that quickly oscillating integrals with sufficiently smooth integrands decay quickly. The most common example of this is its use in showing that the decay of function's Fourier transform depends on the smoothness of that function, as described below.
Many of the following antiderivatives have a term of the form ln |ax + b|.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]
A technical issue in Lebesgue integration is that the domain of integration is defined as a set (a subset of a measure space), with no notion of orientation. In elementary calculus, one defines integration with respect to an orientation : ∫ b a f = − ∫ a b f . {\displaystyle \int _{b}^{a}f=-\int _{a}^{b}f.}
In mathematics, a nonelementary antiderivative of a given elementary function is an antiderivative (or indefinite integral) that is, itself, not an elementary function. [1] A theorem by Liouville in 1835 provided the first proof that nonelementary antiderivatives exist. [2]
The following three basic theorems on the interchange of limits are essentially equivalent: the interchange of a derivative and an integral (differentiation under the integral sign; i.e., Leibniz integral rule); the change of order of partial derivatives; the change of order of integration (integration under the integral sign; i.e., Fubini's ...