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
- <math>
\int_{\phi(a)}^{\phi(b)} f(x)\,dx = \int_{a}^{b} f(\phi(t)) \phi'(t)\,dt
</math>
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.
(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.
Examples
Consider the integral
- <math>
\int_{0}^2 t \cos(t^2+1) \,dt
</math>
By using the substitution
x =
t^{2} + 1, we obtain
dx = 2
t dt and
- <math>
\int_{0}^2 t \cos(t^2+1) \,dt = \frac{1}{2} \int_{0}^2 \cos(t^2+1) 2t \,dt = \frac{1}{2} \int_{1}^{5}\cos(x)\,dx = \frac{1}{2}(\sin(5)-\sin(1)).
</math>
Here we used the substitution rule "from right to left". Note how the lower limit
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
- <math>
\int_0^1 \sqrt{1-x^2}\; dx
</math>
the formula needs to be used from left to right:
the substitution
x = sin(
t),
dx = cos(
t)
dt is useful, because √(1-sin
^{2}(
t)) = cos(
t):
- <math>
\int_0^1 \sqrt{1-x^2}\; dx = \int_0^\pi \sqrt{1-\sin^2(t)} \cos(t)\;dt = \int_0^\pi \cos^2(t)\;dt
</math>
The resulting integral can be computed using
integration by parts.
Antiderivatives
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:
- <math>\int t \cos(t^2+1) \,dt = \frac{1}{2} \int \cos(t^2+1) 2t \,dt </math>
- <math>\qquad = \frac{1}{2} \int\cos(x)\,dx = \frac{1}{2}\sin(x) + C = \frac{1}{2}\sin(t^2+1) + C.
</math>
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
- <math>dx_1\cdots dx_n = |\det(D\phi)| \, dt_1\cdots dt_n</math>
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.
- Give precise statement and example of multivariable substitution; generalization to measure spaces
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