Substitution for single variable
Relation to the fundamental theorem of calculusLet I ⊆ ℝ be an interval and ϕ : [a,b] → I be a continuously differentiable function. Suppose that ƒ : I → ℝ is a continuous function. Then
The formula is used to transform one integral into another integral that is easier to compute. 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 or w-substitution.
Integration by substitution can be derived from the fundamental theorem of calculus as follows. Let ƒ and ϕ be two functions satisfying the above hypothesis that ƒ is continuous on I and ϕ′ is continuous on the closed interval [a,b]. Then the function ƒ(ϕ(t))ϕ′(t) is also continuous on [a,b]. Hence the integrals
Since ƒ is continuous, it possesses an antiderivative F. The composite function F∘ϕ is then defined. Since F and ϕ are differentiable, the chain rule gives
ExamplesConsider the integral
(1) Definite integral
For the integral
Substitution can be used to determine antiderivatives. One chooses a relation between x and u, determines the corresponding relation between dx and du by differentiating, and performs the substitutions. An antiderivative for the substituted function can hopefully be determined; the original substitution between u and x is then undone.
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 u = x2 + 1.
Substitution for multiple variablesOne may also use substitution when integrating functions of several variables. Here the substitution function (v1,...,vn) = φ(u1, ..., un ) needs to be injective and continuously differentiable, and the differentials transform as
More precisely, the change of variables formula is stated in the next theorem:
Theorem. Let U be an open set in Rn and φ : U → Rn an injective differentiable function with continuous partial derivatives, the Jacobian of which is nonzero for every x in U. Then for any real-valued, compactly supported, continuous function f, with support contained in φ(U),
For Lebesgue measurable functions, the theorem can be stated in the following form (Fremlin 2010, Theorem 263D):
Theorem. Let U be a measurable subset of Rn and φ : U → Rn an injective function, and suppose for every x in U there exists φ'(x) in Rn,n such that φ(y) = φ(x) + φ'(x) (y − x) + o(||y − x||) as y → x. Then φ(U) is measurable, and for any real-valued function f defined on φ(U),
Another very general version in measure theory is the following (Hewitt & Stromberg 1965, Theorem 20.3):
Theorem. Let X be a locally compact Hausdorff space equipped with a finite Radon measure μ, and let Y be a σ-compact Hausdorff space with a σ-finite Radon measure ρ. Let φ : X → Y be a continuous and absolutely continuous function (where the latter means that ρ(φ(E)) = 0 whenever μ(E) = 0). Then there exists a real-valued Borel measurable function w on X such that for every Lebesgue integrable function f : Y → R, the function (f φ)w is Lebesgue integrable on X, and
In geometric measure theory, integration by substitution is used with Lipschitz functions. A bi-Lipschitz function is a Lipschitz function φ : U → Rn which is one-to-one, and such that its inverse function φ−1 : φ(U) → U is also Lipschitz. By Rademacher's theorem a bi-Lipschitz mapping is differentiable almost everywhere. In particular, the Jacobian determinant of a bi-Lipschitz mapping det D'φ is well-defined almost everywhere. The following result then holds:
Theorem. Let U be an open subset of Rn and φ : U → Rn be a bi-Lipschitz mapping. Let f : φ(U) → R be measurable. Then
Application in probabilitySubstitution can be used to answer the following important question in probability: given a random variable with probability density and another random variable related to by the equation , what is the probability density for ?
It is easiest to answer this question by first answering a slightly different question: what is the probability that takes a value in some particular subset ? Denote this probability . Of course, if has probability density then the answer is
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