Substitution rule

In calculus, the substitution rule is a tool for finding antiderivatives and integrals. Using the fundamental theorem of calculus often requires finding an antiderivative. For this and other reasons, the substitution rule is a relatively important tool for mathematicians. It is the counterpart to the chain rule of differentiation.

Related Topics:
Calculus - Antiderivative - Integral - Fundamental theorem of calculus - Chain rule - Differentiation

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Suppose f(x) is an integrable function, and φ(t)  is a continuously differentiable function which is defined on the interval and whose image is contained in the domain of f. Then

Related Topics:
Function - Interval - Image - Domain

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:

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int_{phi(a)}^{phi(b)} f(x),dx = int_{a}^{b} f(phi(t)) phi'(t),dt

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The formula is best remembered using Leibniz' formalism: the substitution x = φ(t)  yields dx/dt = φ'(t)  and thus formally dx = φ'(t)dt , which is precisely the required substitution for dx.

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(In fact, one may view the substitution rule as a major justification of the Leibniz formalism for integrals and derivatives.)

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The formula is used to transform an integral into another one which (hopefully) is easier to determine. Thus, the formula can be used "from left to right" or "from right to left" in order to simplify a given integral; when used in the latter manner, it is sometimes known as u-substitution.

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