In calculus, integration by substitution, also known as u-substitution, reverse chain rule or change of variables,[1] is a method for evaluating integrals and antiderivatives.
Before stating the result rigorously, consider a simple case using indefinite integrals.
This procedure is frequently used, but not all integrals are of a form that permits its use.
In any event, the result should be verified by differentiating and comparing to the original integrand.
Working heuristically with infinitesimals yields the equation
(This equation may be put on a rigorous foundation by interpreting it as a statement about differential forms.)
Integration by substitution can be derived from the fundamental theorem of calculus as follows.
in fact exist, and it remains to show that they are equal.
is differentiable, combining the chain rule and the definition of an antiderivative gives:
The tangent function can be integrated using substitution by expressing it in terms of the sine and cosine:
The cotangent function can be integrated similarly by expressing it as
When evaluating definite integrals by substitution, one may calculate the antiderivative fully first, then apply the boundary conditions.
In that case, there is no need to transform the boundary terms.
Alternatively, one may fully evaluate the indefinite integral (see above) first then apply the boundary conditions.
where det(Dφ)(u1, ..., un) denotes the determinant of the Jacobian matrix of partial derivatives of φ at the point (u1, ..., un).
This formula expresses the fact that the absolute value of the determinant of a matrix equals the volume of the parallelotope spanned by its columns or rows.
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.
First, the requirement that φ be continuously differentiable can be replaced by the weaker assumption that φ be merely differentiable and have a continuous inverse.
[4] This is guaranteed to hold if φ is continuously differentiable by the inverse function theorem.
Alternatively, the requirement that det(Dφ) ≠ 0 can be eliminated by applying Sard's theorem.
[5] For Lebesgue measurable functions, the theorem can be stated in the following form:[6] 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 (here o is little-o notation).
in the sense that if either integral exists (including the possibility of being properly infinite), then so does the other one, and they have the same value.
Another very general version in measure theory is the following:[7] 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.
In geometric measure theory, integration by substitution is used with Lipschitz functions.
By Rademacher's theorem, a bi-Lipschitz mapping is differentiable almost everywhere.
In particular, the Jacobian determinant of a bi-Lipschitz mapping det Dφ is well-defined almost everywhere.
in the sense that if either integral exists (or is properly infinite), then so does the other one, and they have the same value.
The above theorem was first proposed by Euler when he developed the notion of double integrals in 1769.
Although generalized to triple integrals by Lagrange in 1773, and used by Legendre, Laplace, and Gauss, and first generalized to n variables by Mikhail Ostrogradsky in 1836, it resisted a fully rigorous formal proof for a surprisingly long time, and was first satisfactorily resolved 125 years later, by Élie Cartan in a series of papers beginning in the mid-1890s.
[8][9] Substitution can be used to answer the following important question in probability: given a random variable X with probability density pX and another random variable Y such that Y= ϕ(X) for injective (one-to-one) ϕ, what is the probability density for Y?