In calculus, the

**substitution rule**is an important tool for finding antiderivatives and integrals. It is the counterpart to the chain rule for differentiation.

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

*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.

Table of contents |

2 Antiderivatives 3 Substitution rule for multiple variables |

## Examples

By using the substitution*x*=

*t*

^{2}+ 1, we obtain

*dx*= 2

*t*

*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*):

## Antiderivatives

Similar 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 substitution*x*=

*t*

^{2}+ 1.

## Substitution rule for multiple variables

One may also use substitution when integrating functions of several variables.
Here the substitution function (*x*_{1},...,*x*_{n}) = φ(*t*_{1},...,*t*_{n}) needs to be one-to-one and continuously differentiable, and the differentials transform as

*Give precise statement and example of multivariable substitution; generalization to measure spaces*