Differentiation under the integral sign
Encyclopedia
Differentiation under the integral sign is a useful operation in calculus
. Suppose that it is required to differentiate with respect to x the function
where the functions
and 
are both continuous in both 
and 
in some region of the 
plane, including 
, 
, and the functions 
and 
are both continuous and both have continuous derivatives for 
. Then for 
:
This formula is the general form of the Leibniz integral rule
and can be derived using the
fundamental theorem of calculus
. The fundamental theorem of calculus is just a particular case of the above formula, for
, a constant, 
and 
.
If both upper and lower limits are taken as constants, then the formula takes the shape of an operator equation:
where
is the partial derivative
with respect to
and 
is the integral operator with respect to 
over a fixed interval
. That is, it is related to the symmetry of second derivatives
, but involving integrals as well as derivatives. This case is also known as the Leibniz integral rule
.
The following three basic theorems on the interchange of limits
are essentially equivalent:
as the Reynolds transport theorem
:
where
is a scalar function, 
and 
denote a time-varying connected region of 
and its boundary, respectively, 
is the Eulerian velocity of the boundary (see Lagrangian and Eulerian coordinates
) and
is unit outwards normal.
The general statement of the Leibniz integral rule requires concepts from differential geometry
, specifically differential forms, exterior derivative
s, wedge products and interior products. With those tools, the Leibniz integral rule in
-dimensions is:
where
is a time-varying domain of integration, 
is a 
-form, 
is the vector field of the velocity, 
, 
denotes the interior product, 
is the exterior derivative
of
with respect to the space variables only and 
is the time-derivative of 
and its lower limit 
If
is a continuous function
of
or 
, then, from the definition of the definite integral, 
and 
since, from proof of the fundamental theorem of calculus,
Suppose
and 
are constant, and that 
involves a parameter 
which is constant in the integration but may vary to form different integrals. Then, by the definition of a function,
In general, this may be differentiated by differentiating under the integral sign; i.e.,
To prove this and, at the same time, to determine conditions under which the formula is true, we proceed as follows:
From
From the fact that
we have
If
is a continuous function of 
and 
when 
then for any 
there exists 
such that
for all values of 
(This follows from the Heine–Cantor theorem
that every continuous function on a compact set is uniformly continuous.)
Therefore, from
and 
we get
and the fact that
is, therefore, a continuous function.
Similarly if
exists and is continuous, then for all 
there exists 
such that:
for all 
Therefore,
Now,
as 
, therefore,
This is the formula we set out to prove.
Now, suppose
where 
and 
are functions of 
which take increments 
and 
respectively, when 
is increased by 
Then,
A form of the mean value theorem
,
where 
can be applied to the first and last integrals of the formula for 
above, resulting in
Dividing by
, letting 
noticing 
and 
and using the result 
yields
This is the general form of the Leibniz integral rule
.
If
If
The function
is not continuous at the point 
and the function 
has a discontinuity at 
because 
approaches 
as 
and approaches 
as 
If we now differentiate
with respect to 
under the integral sign, we get
,
which is, of course, true for all values of
except 
Consider integrating
(for
).
Now,
As
varies from 
to 
varies through positive values from 
to 
when 
and 
varies through negative values from 
to 
when 
or 
Hence,
when 
and
when 
or 
Therefore,
when 
and
when 
or 
Upon integrating both sides with respect to
, we get 
when 
and 
when 
or 
may be determined by setting 
in
Thus,
Hence, 
when 
To determine
in the same manner, we should need to substitute in 
a value of 
greater numerically than 1. This is somewhat inconvenient. Instead, we substitute 
, where 
Then,
Therefore,
(and 
when 
or 
)
The definition of
is now complete:
when 
and
when 
or 
The foregoing discussion, of course, does not apply when
since the conditions for differentiability are not met.
where both
, by differentiating under the integral sign.
Let us first find
Dividing both the numerator and the denominator by
yields
The limits of integration being independent of
gives us
whereas
gives us
Equating these two relations then yields
In a similar fashion, pursuing
yields
Adding the two results then produces
which is the value of the integral
Note that if we define
it can easily be shown that
Given
this partial-derivative-based recursive relation (i.e., integral reduction formula) can then be utilized to compute all of the values of 
for 
(
, 
, 
etc.).
for
Differentiating under the integral with respect to
we have
Now, when
, we have, from 
Hence,
which is the value of the integral
We introduce a new variable
and rewrite the integral as
Note that for
Thus, we proceed
From the equation for
we can see 
So, integrating both sides of 
with respect to 
between the limits 
and 
yields
which is the value of the integral
. In this example, we shall simply apply the above given formula, to get
where the derivative with respect to x of hyperbolic cosine t squared is 0. This is a simple example on how to use this formula for variable limits.
and for
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
. Suppose that it is required to differentiate with respect to x the function
where the functions
and are both continuous in both and in some region of the plane, including , , and the functions and are both continuous and both have continuous derivatives for . Then for :This formula is the general form of the Leibniz integral rule
Leibniz integral rule
In mathematics, Leibniz's rule for differentiation under the integral sign, named after Gottfried Leibniz, tells us that if we have an integral of the formthen for x \in the derivative of this integral is thus expressible...
and can be derived using the
fundamental theorem of calculus
Fundamental theorem of calculus
The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation...
. The fundamental theorem of calculus is just a particular case of the above formula, for
, a constant, and .If both upper and lower limits are taken as constants, then the formula takes the shape of an operator equation:
where
is the partial derivativePartial 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...
with respect to
and is the integral operator with respect to over a fixed intervalInterval
Interval may refer to:* Interval , a range of numbers * Interval measurements or interval variables in statistics is a level of measurement...
. That is, it is related to the symmetry of second derivatives
Symmetry of second derivatives
In mathematics, the symmetry of second derivatives refers to the possibility of interchanging the order of taking partial derivatives of a functionfof n variables...
, but involving integrals as well as derivatives. This case is also known as the Leibniz integral rule
Leibniz integral rule
In mathematics, Leibniz's rule for differentiation under the integral sign, named after Gottfried Leibniz, tells us that if we have an integral of the formthen for x \in the derivative of this integral is thus expressible...
.
The following three basic theorems on the interchange of limits
Interchange of limiting operations
In mathematics, the study of interchange of limiting operations is one of the major concerns of mathematical analysis. in that two given limiting operations, say L and M, cannot be assumed to give the same result when applied in either order...
are essentially equivalent:
- the interchange of a derivative and an integral (differentiation under the integral sign; i.e., Leibniz integral ruleLeibniz integral ruleIn mathematics, Leibniz's rule for differentiation under the integral sign, named after Gottfried Leibniz, tells us that if we have an integral of the formthen for x \in the derivative of this integral is thus expressible...
) - the change of order of partial derivatives
- the change of order of integration (integration under the integral sign; i.e., Fubini's theoremFubini's theoremIn mathematical analysis Fubini's theorem, named after Guido Fubini, is a result which gives conditions under which it is possible to compute a double integral using iterated integrals. As a consequence it allows the order of integration to be changed in iterated integrals.-Theorem...
)
Higher dimensions
The Leibniz integral rule can be extended to multidimensional integrals. In two and three dimensions, this rule is better known from the field of fluid dynamicsFluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
as the Reynolds transport theorem
Reynolds transport theorem
Reynolds' transport theorem , or in short Reynolds theorem, is a three-dimensional generalization of the Leibniz integral rule which is also known as differentiation under the integral sign....
:
where
is a scalar function, and denote a time-varying connected region of and its boundary, respectively, is the Eulerian velocity of the boundary (see Lagrangian and Eulerian coordinatesLagrangian and Eulerian coordinates
In fluid dynamics and finite-deformation plasticity the Lagrangian specification of the flow field is a way of looking at fluid motion where the observer follows an individual fluid parcel as it moves through space and time. Plotting the position of an individual parcel through time gives the...
) and
is unit outwards normal.The general statement of the Leibniz integral rule requires concepts from differential geometry
Differential geometry and topology
Differential geometry is a mathematical discipline that uses the techniques of differential and integral calculus, as well as linear and multilinear algebra, to study problems in geometry. The theory of plane and space curves and of surfaces in the three-dimensional Euclidean space formed the basis...
, specifically differential forms, exterior derivative
Exterior derivative
In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....
s, wedge products and interior products. With those tools, the Leibniz integral rule in
-dimensions is:where
is a time-varying domain of integration, is a -form, is the vector field of the velocity, , denotes the interior product, is the exterior derivativeExterior derivative
In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....
of
with respect to the space variables only and is the time-derivative of Derivation of the principle of differentiation under the integral sign
A definite integral is a function of its upper limit and its lower limit If
is a continuous functionFunction (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
of
or , then, from the definition of the definite integral, and since, from proof of the fundamental theorem of calculus,
Suppose
and are constant, and that involves a parameter which is constant in the integration but may vary to form different integrals. Then, by the definition of a function,In general, this may be differentiated by differentiating under the integral sign; i.e.,
To prove this and, at the same time, to determine conditions under which the formula is true, we proceed as follows:
From
From the fact that
we haveIf
is a continuous function of and when then for any there exists such that for all values of (This follows from the Heine–Cantor theorem
Heine–Cantor theorem
In mathematics, the Heine–Cantor theorem, named after Eduard Heine and Georg Cantor, states that if M and N are metric spaces and M is compact then every continuous functionis uniformly continuous....
that every continuous function on a compact set is uniformly continuous.)
Therefore, from
and we get
and the fact that
is, therefore, a continuous function.Similarly if
exists and is continuous, then for all there exists such that: for all Therefore,
Now,
as , therefore,This is the formula we set out to prove.
Now, suppose
where and are functions of which take increments and respectively, when is increased by Then,A form of the mean value theorem
Mean value theorem
In calculus, the mean value theorem states, roughly, that given an arc of a differentiable curve, there is at least one point on that arc at which the derivative of the curve is equal to the "average" derivative of the arc. Briefly, a suitable infinitesimal element of the arc is parallel to the...
,
where can be applied to the first and last integrals of the formula for above, resulting inDividing by
, letting noticing and and using the result yieldsThis is the general form of the Leibniz integral rule
Leibniz integral rule
In mathematics, Leibniz's rule for differentiation under the integral sign, named after Gottfried Leibniz, tells us that if we have an integral of the formthen for x \in the derivative of this integral is thus expressible...
.
Example 1
If
If
The function
is not continuous at the point and the function has a discontinuity at because approaches as and approaches as If we now differentiate
with respect to under the integral sign, we get,which is, of course, true for all values of
except Example 2
The principle of differentiating under the integral sign may sometimes be used to evaluate a definite integral.Consider integrating
(for
).Now,
As
varies from to varies through positive values from to when and varies through negative values from to when or Hence,
when and
when or Therefore,
when and when or Upon integrating both sides with respect to
, we get when and when or may be determined by setting inThus,
Hence, when To determine
in the same manner, we should need to substitute in a value of greater numerically than 1. This is somewhat inconvenient. Instead, we substitute , where Then,Therefore,
(and when or )The definition of
is now complete: when and when or The foregoing discussion, of course, does not apply when
since the conditions for differentiability are not met.Example 3
Here, we consider the integration ofwhere both
, by differentiating under the integral sign.Let us first find
Dividing both the numerator and the denominator by
yieldsThe limits of integration being independent of
gives uswhereas
gives usEquating these two relations then yields
In a similar fashion, pursuing
yieldsAdding the two results then produces
which is the value of the integral
Note that if we define
it can easily be shown that
Given
this partial-derivative-based recursive relation (i.e., integral reduction formula) can then be utilized to compute all of the values of for (, , etc.).Example 4
Here, we consider the integralfor
Differentiating under the integral with respect to
we haveNow, when
, we have, from Hence,
which is the value of the integral
Example 5
Here, we consider the integralWe introduce a new variable
and rewrite the integral asNote that for
Thus, we proceed
From the equation for
we can see So, integrating both sides of with respect to between the limits and yieldswhich is the value of the integral
Example 6
Here is an example that has variable limits. Let us try to find. In this example, we shall simply apply the above given formula, to getwhere the derivative with respect to x of hyperbolic cosine t squared is 0. This is a simple example on how to use this formula for variable limits.
Other problems
There are innumerable other integrals that can be solved "quickly" using the technique of differentiation under the integral sign. For examples, to solveand for
Popular culture
- Differentiation under the integral sign is mentioned in the late physicistPhysicistA physicist is a scientist who studies or practices physics. Physicists study a wide range of physical phenomena in many branches of physics spanning all length scales: from sub-atomic particles of which all ordinary matter is made to the behavior of the material Universe as a whole...
Richard Feynman'sRichard FeynmanRichard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...
best-selling memoir Surely You're Joking, Mr. Feynman!Surely You're Joking, Mr. Feynman!"Surely You're Joking, Mr. Feynman!": Adventures of a Curious Character is an edited collection of reminiscences by the Nobel Prize-winning physicist Richard Feynman. The book, released in 1985, covers a variety of instances in Feynman's life...
(in the chapter "A Different Box of Tools"), where he mentions learning it from an old text, Advanced Calculus (1926), by Frederick S. Woods (who was a professor of mathematics in the Massachusetts Institute of TechnologyMassachusetts Institute of TechnologyThe Massachusetts Institute of Technology is a private research university located in Cambridge, Massachusetts. MIT has five schools and one college, containing a total of 32 academic departments, with a strong emphasis on scientific and technological education and research.Founded in 1861 in...
) while in high schoolHigh schoolHigh school is a term used in parts of the English speaking world to describe institutions which provide all or part of secondary education. The term is often incorporated into the name of such institutions....
. The technique was not often taught when Feynman later received his formal education in calculusCalculusCalculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
and, knowing it, Feynman was able to use the technique to solve some otherwise difficult integration problems upon his arrival at graduate school at Princeton UniversityPrinceton UniversityPrinceton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
. The direct citation from Surely You're Joking, Mr. Feynman! regarding the method of differentiation under the integral sign is as follows: