In
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ... , integration by substitution, also known as ''u''-substitution, reverse chain rule or change of variables, is a method for evaluating
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ... s and
antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolica ... s. It is the counterpart to the
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ... for
differentiation , and can loosely be thought of as using the chain rule "backwards".
Substitution for a single variable
Introduction
Before stating the result
rigorously , consider a simple case using
indefinite integral
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolicall ... s.
Compute
\textstyle\int(2x^3+1)^7(x^2)\,dx .
Set
u=2x^3+1 . This means
\textstyle\frac=6x^2 , or in
differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many application ... ,
du=6x^2\,dx . Now
:
\int(2x^3 +1)^7(x^2)\,dx = \frac\int\underbrace_\underbrace_=\frac\int u^\,du=\frac\left(\fracu^\right)+C=\frac(2x^3+1)^+C,
where
C is an arbitrary
constant of integration
In calculus, the constant of integration, often denoted by C (or c), is a constant term added to an antiderivative of a function f(x) to indicate that the indefinite integral of f(x) (i.e., the set of all antiderivatives of f(x)), on a connecte ... .
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.
:
\frac\left frac(2x^3+1)^+C\right \frac(2x^3+1)^(6x^2) = (2x^3+1)^7(x^2).
For definite integrals, the limits of integration must also be adjusted, but the procedure is mostly the same.
Definite integrals
Let
g: ,b rightarrow I be a
differentiable function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ... with a
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ... derivative, where
I \subset \mathbb is an
interval . Suppose that
f:I\rightarrow\mathbb is a
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ... . Then
:
\int_a^b f(g(x))\cdot g'(x)\, dx = \int_^ f(u)\ du.
In Leibniz notation, the substitution
u=g(x) yields
:
\frac = g'(x).
Working heuristically with
infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally re ... s yields the equation
:
du = g'(x)\,dx,
which suggests the substitution formula above. (This equation may be put on a rigorous foundation by interpreting it as a statement about
differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many application ... s.) One may view the method of integration by substitution as a partial justification of
Leibniz's notation
In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols and to represent infinitely small (or infinitesimal) increments of and , respectively, just a ... for integrals and derivatives.
The formula is used to transform one integral into another integral that is easier to compute. Thus, the formula can be read from left to right or from right to left in order to simplify a given integral. When used in the former manner, it is sometimes known as ''u''-substitution or ''w''-substitution in which a new variable is defined to be a function of the original variable found inside the
composite
Composite or compositing may refer to:
Materials
* Composite material, a material that is made from several different substances
** Metal matrix composite, composed of metal and other parts
** Cermet, a composite of ceramic and metallic materials ... function multiplied by the derivative of the inner function. The latter manner is commonly used in
trigonometric substitution , replacing the original variable with a
trigonometric function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ... of a new variable and the original
differential with the differential of the trigonometric function.
Proof
Integration by substitution can be derived from the
fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, ... as follows. Let
f and
g be two functions satisfying the above hypothesis that
f is continuous on
I and
g' is integrable on the closed interval
,b /math>. Then the function f(g(x))\cdot g'(x) is also integrable on ,b /math>. Hence the integrals
:\int_a^b f(g(x))\cdot g'(x)\ dx
and
:\int_^ f(u)\ du
in fact exist, and it remains to show that they are equal.
Since f is continuous, it has an antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolica ... F . The composite function
In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ... F \circ g is then defined. Since g is differentiable, combining the chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ... and the definition of an antiderivative gives
:(F \circ g)'(x) = F'(g(x)) \cdot g'(x) = f(g(x)) \cdot g'(x).
Applying the fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, ... twice gives
:
\begin
\int_a^b f(g(x)) \cdot g'(x)\ dx
&= \int_a^b (F \circ g)'(x)\ dx \\
&= (F \circ g)(b) - (F \circ g)(a) \\
&= F(g(b)) - F(g(a)) \\
&= \int_^ f(u)\ du,
\end
which is the substitution rule.
Examples
Example 1
Consider the integral
:\int_0^2 x \cos(x^2+1)\ dx.
Make the substitution u = x^ + 1 to obtain du = 2x\ dx , meaning x\ dx = \frac\ du . Therefore,
:\begin
\int_^ x \cos(x^2+1) \ dx
&= \frac \int_^\cos(u)\ du \\ pt &= \frac(\sin(5)-\sin(1)).
\end
Since the lower limit x = 0 was replaced with u = 1 , and the upper limit x = 2 with 2^ + 1 = 5 , a transformation back into terms of x was unnecessary.
Alternatively, one may fully evaluate the indefinite integral ( see below ) first then apply the boundary conditions. This becomes especially handy when multiple substitutions are used.
Example 2
For the integral
:\int_0^1 \sqrt\,dx,
a variation of the above procedure is needed. The substitution x = \sin u implying dx = \cos u \,du is useful because \sqrt = \cos(u) . We thus have
:\begin
\int_0^1 \sqrt\ dx
&= \int_0^ \sqrt \cos(u)\ du \\ pt &= \int_0^ \cos^2u\ du \\ pt &= \left frac + \frac\right 0^ \\ pt &= \frac + 0 \\
&= \frac.
\end
The resulting integral can be computed using integration by parts
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivat ... or a double angle formula
In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ... , 2\cos^ u = 1 + \cos (2u) , followed by one more substitution. One can also note that the function being integrated is the upper right quarter of a circle with a radius of one, and hence integrating the upper right quarter from zero to one is the geometric equivalent to the area of one quarter of the unit circle, or \frac\pi 4 .
Antiderivatives
Substitution can be used to determine antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolica ... s. 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 x and u is then undone.
Similar to example 1 above, the following antiderivative can be obtained with this method:
:\begin
\int x \cos(x^2+1) \,dx
&= \frac \int 2x \cos(x^2+1) \,dx \\ pt &= \frac \int\cos u\,du \\ pt &= \frac\sin u + C \\
&= \frac\sin(x^2+1) + C,
\end
where C is an arbitrary constant of integration
In calculus, the constant of integration, often denoted by C (or c), is a constant term added to an antiderivative of a function f(x) to indicate that the indefinite integral of f(x) (i.e., the set of all antiderivatives of f(x)), on a connecte ... .
There were no integral boundaries to transform, but in the last step reverting the original substitution u = x^ + 1 was necessary. 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.
The tangent function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ... can be integrated using substitution by expressing it in terms of the sine and cosine:
:\int \tan x \,dx = \int \frac \,dx
Using the substitution u = \cos x gives du = -\sin x\,dx and
:\begin
\int \tan x \,dx &= \int \frac \,dx \\
&= \int -\frac \\
&= -\ln , u, + C \\
&= -\ln , \cos x, + C \\
&= \ln , \sec x, + C.
\end
Substitution for multiple variables
One may also use substitution when integrating functions of several variables .
Here the substitution function needs to be injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ... and continuously differentiable, and the differentials transform as
:dv_1 \cdots dv_n = \left, \det(D\varphi)(u_1, \ldots, u_n)\ \, du_1 \cdots du_n,
where denotes the determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ... of the Jacobian matrix
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variable ... of partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ... s of at the point . This formula expresses the fact that the absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ... of the determinant of a matrix equals the volume of the parallelotope spanned by its columns or rows.
More precisely, the ''change of variables
Change or Changing may refer to:
Alteration
* Impermanence, a difference in a state of affairs at different points in time
* Menopause, also referred to as "the change", the permanent cessation of the menstrual period
* Metamorphosis, or change, ... '' formula is stated in the next theorem:
Theorem. Let be an open set in and an injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ... differentiable function with continuous partial derivatives, the Jacobian of which is nonzero for every in . Then for any real-valued, compactly supported, continuous function , with support contained in ,
:\int_ f(\mathbf)\, d\mathbf
= \int_U f(\varphi(\mathbf)) \left, \det(D\varphi)(\mathbf)\ \,d\mathbf.
The conditions on the theorem can be weakened in various ways. First, the requirement that be continuously differentiable can be replaced by the weaker assumption that be merely differentiable and have a continuous inverse. This is guaranteed to hold if is continuously differentiable by the inverse function theorem . Alternatively, the requirement that can be eliminated by applying Sard's theorem
In mathematics, Sard's theorem, also known as Sard's lemma or the Morse–Sard theorem, is a result in mathematical analysis that asserts that the set of critical values (that is, the image of the set of critical points) of a smooth function ... .
For Lebesgue measurable functions, the theorem can be stated in the following form:
Theorem. Let be a measurable subset of and an injective function
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ... , and suppose for every in there exists in such that as (here is little-''o'' notation ). Then is measurable, and for any real-valued function defined on ,
:\int_ f(v)\, dv = \int_U f(\varphi(u)) \left, \det \varphi'(u)\ \,du
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
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ... is the following:
Theorem. Let be a locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ... Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the ma ... equipped with a finite Radon measure
In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all Borel ... , and let be a σ-compact Hausdorff space with a σ-finite Radon measure . Let be an absolutely continuous
In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central ope ... function (where the latter means that whenever ). Then there exists a real-valued Borel measurable function on such that for every Lebesgue integrable
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Leb ... function , the function is Lebesgue integrable on , and
:\int_Y f(y)\,d\rho(y) = \int_X (f\circ \varphi)(x)\,w(x)\,d\mu(x).
Furthermore, it is possible to write
:w(x) = (g\circ \varphi)(x)
for some Borel measurable function on .
In geometric measure theory
In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians to extend tools from differential geometry to a much larger class of surfa ... , integration by substitution is used with Lipschitz function
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there e ... s. A bi-Lipschitz function is a Lipschitz function which is injective and whose inverse function is also Lipschitz. By Rademacher's theorem In mathematical analysis, Rademacher's theorem, named after Hans Rademacher, states the following: If is an open subset of and is Lipschitz continuous, then is differentiable almost everywhere in ; that is, the points in at which is ''not'' di ... a bi-Lipschitz mapping is differentiable almost everywhere
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ... . In particular, the Jacobian determinant of a bi-Lipschitz mapping is well-defined almost everywhere. The following result then holds:
Theorem. Let be an open subset of and be a bi-Lipschitz mapping. Let be measurable. Then
:\int_U (f\circ \varphi)(x) , \det D\varphi(x), \,dx = \int_ f(x)\,dx
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
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ... when he developed the notion of double integrals in 1769. Although generalized to triple integrals by Lagrange
Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia[Legendre, ](_blank)Laplace
Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ... , Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ... , and first generalized to variables by Mikhail Ostrogradski 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
Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometr ... in a series of papers beginning in the mid-1890s.
Application in probability
Substitution can be used to answer the following important question in probability: given a random variable X with probability density p_X and another random variable Y such that Y=\phi(X) for injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ... (one-to-one) \phi , what is the probability density for Y ?
It is easiest to answer this question by first answering a slightly different question: what is the probability that Y takes a value in some particular subset S ? Denote this probability P(Y \in S) . Of course, if Y has probability density p_Y then the answer is
:P(Y \in S) = \int_S p_Y(y)\,dy,
but this isn't really useful because we don't know p_Y ; it's what we're trying to find. We can make progress by considering the problem in the variable X . Y takes a value in S whenever X takes a value in \phi^(S) , so
:P(Y \in S) = P(X \in \phi^(S)) = \int_ p_X(x)\,dx.
Changing from variable x to y gives
:P(Y \in S) = \int_ p_X(x)\,dx = \int_S p_X(\phi^(y)) \left, \frac\\,dy.
Combining this with our first equation gives
:\int_S p_Y(y)\,dy = \int_S p_X(\phi^(y)) \left, \frac\\,dy,
so
:p_Y(y) = p_X(\phi^(y)) \left, \frac\.
In the case where X and Y depend on several uncorrelated variables, i.e. p_X=p_X(x_1, \ldots, x_n) and y=\phi(x) , p_Y can be found by substitution in several variables discussed above. The result is
:p_Y(y) = p_X(\phi^(y)) \left, \det D\phi ^(y) \.
See also
*Probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
*Substitution of variables
Substitution may refer to:
Arts and media
*Chord substitution, in music, swapping one chord for a related one within a chord progression
*Substitution (poetry), a variation in poetic scansion
* "Substitution" (song), a 2009 song by Silversun Pic ...
* Trigonometric substitution
*Weierstrass substitution
In integral calculus, the tangent half-angle substitution is a change of variables used for evaluating integrals, which converts a rational function of trigonometric functions of x into an ordinary rational function of t by setting t = \tan \tf ...
* Euler substitution
* Glasser's master theorem
*Pushforward measure In measure theory, a pushforward measure (also known as push forward, push-forward or image measure) is obtained by transferring ("pushing forward") a measure from one measurable space to another using a measurable function.
Definition
Given meas ...
Notes
References
*
*
* .
* .
*
* .
*
* .
External links
Integration by substitution at Encyclopedia of Mathematics
The ''Encyclopedia of Mathematics'' (also ''EOM'' and formerly ''Encyclopaedia of Mathematics'') is a large reference work in mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structu ...
Area formula at Encyclopedia of Mathematics
The ''Encyclopedia of Mathematics'' (also ''EOM'' and formerly ''Encyclopaedia of Mathematics'') is a large reference work in mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structu ...
{{Integrals
Articles containing proofs
Integral calculus
es:Métodos de integración#Método de integración por sustitución
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