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In calculus, integration by substitution, also known as u-substitution, reverse chain rule or change of variables, [1] is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation , and can loosely be thought of as using the chain rule "backwards."
For example, suppose we want to find the integral ∫ 0 ∞ x 2 e − 3 x d x . {\displaystyle \int _{0}^{\infty }x^{2}e^{-3x}\,dx.} Since this is a product of two functions that are simple to integrate separately, repeated integration by parts is certainly one way to evaluate it.
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.
Example 1a. The function is f(x, y) = (x − 1) 2 + √ y; if one adopts the substitution u = x − 1, v = y therefore x = u + 1, y = v one obtains the new function f 2 (u, v) = (u) 2 + √ v. Similarly for the domain because it is delimited by the original variables that were transformed before (x and y in example)
Feynman parametrization is a technique for evaluating loop integrals which arise from Feynman diagrams with one or more loops. However, it is sometimes useful in integration in areas of pure mathematics as well.
A line integral (sometimes called a path integral) is an integral where the function to be integrated is evaluated along a curve. [42] Various different line integrals are in use. In the case of a closed curve it is also called a contour integral. The function to be integrated may be a scalar field or a vector field.
The substitution is described in most integral calculus textbooks since the late 19th century, usually without any special name. [5] It is known in Russia as the universal trigonometric substitution, [6] and also known by variant names such as half-tangent substitution or half-angle substitution.
The following is a list of integrals (antiderivative functions) of rational functions. Any rational function can be integrated by partial fraction decomposition of the function into a sum of functions of the form: