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
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 |
Consider the integral
For the integral
Examples
By using the substitution x = t2 + 1, we obtain dx = 2t dt and
Here we used the substitution rule "from right to left". Note how the lower limit t = 0 was transformed into x = 02 + 1 = 1 and the upper limit t = 2 into x = 22 + 1 = 5.
the formula needs to be used from left to right:
the substitution x = sin(t), dx = cos(t) dt is useful, because √(1-sin2(t)) = cos(t):
The resulting integral can be computed using integration by parts.
The substitution rule can be used to determine antiderivatives. One chooses a relation between x and t, determines the corresponding relation between dx and dt by differentiating, and performs the substitutions. An antiderivative for the substituted function can hopefully be determined; the original substitution between x and t is then undone.
Similar to our first example above, we can determine the following antiderivative with this method:
One may also use substitution when integrating functions of several variables.
Here the substitution function (x1,...,xn) = φ(t1,...,tn) needs to be one-to-one and continuously differentiable, and the differentials transform as
Antiderivatives
Note that there were no integral boundaries to transform, but in the last step we had to revert the original substitution x = t2 + 1.Substitution rule for multiple variables
where det(Dφ) denotes the determinant of the Jacobian matrix containing the partial derivatives of φ. This formula expresses the fact that the absolute value of the determinant of given vectors equals the volume of the spanned parallelepiped.