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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."
The intent is that when expressed in new variables, the problem may become simpler, or equivalent to a better understood problem. Change of variables is an operation that is related to substitution. However these are different operations, as can be seen when considering differentiation or integration (integration by substitution). A very simple ...
Just as the definite integral of a positive function of one variable represents the area of the region between the graph of the function and the x-axis, the double integral of a positive function of two variables represents the volume of the region between the surface defined by the function (on the three-dimensional Cartesian plane where z = f(x, y)) and the plane which contains its domain. [1]
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The substitutions of Euler can be generalized by allowing the use of imaginary numbers. For example, in the integral +, the substitution + = + can be used. Extensions to the complex numbers allows us to use every type of Euler substitution regardless of the coefficients on the quadratic.
For example, ↑ 0 is the identity substitution, leaving a term unchanged. A finite list of terms M 1.M 2...M n abbreviates the substitution M 1.M 2...M n.(n+1).(n+2)... leaving all variables greater than n unchanged. The application of a substitution s to a term M is written M[s]. The composition of two substitutions s 1 and s 2 is written s 1 ...
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.
Two substitutions are considered equal if they map each variable to syntactically equal result terms, formally: σ = τ if xσ = xτ for each variable x ∈ V. The composition of two substitutions σ = { x 1 ↦ t 1, …, x k ↦ t k} and τ = { y 1 ↦ u 1, …, y l ↦ u l} is obtained by removing from the substitution { x 1 ↦ t 1 τ ...
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