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## Substitution ruleIn calculus, thesubstitution rule is an important tool for finding antiderivatives and integrals. It is the counterpart to the chain rule for differentiation.
Suppose
x = φ(t) yields dx/dt = φ'(t) and thus formally dx = φ'(t) dt, which is precisely the required substitution for dx.
(In fact, one may view the substitution rule as a major justification of the Leibniz formalism for integrals and derivatives.)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.
## ExamplesBy using the substitutionx = t^{2} + 1, we obtain dx = 2t dt and
t = 0 was transformed into x = 0^{2} + 1 = 1 and the upper limit t = 2 into x = 2^{2} + 1 = 5. For the integral x = sin(t), dx = cos(t) dt is useful, because √(1-sin^{2}(t)) = cos(t):
## AntiderivativesSimilar to our first example above, we can determine the following antiderivative with this method: Note that there were no integral boundaries to transform, but in the last step we had to revert the original substitutionx = t^{2} + 1.## Substitution rule for multiple variables
One may also use substitution when integrating functions of several variables.
Here the substitution function (
*Give precise statement and example of multivariable substitution; generalization to measure spaces*
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