Differentiation under the integral sign

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Differentiation under the integral sign is a useful operation in calculus. Formally it can be stated as follows:

Theorem. Let f(x, t) be a function such that both f(x, t) and its partial derivative fx(x, t) are continuous in t and x in some region of the (x, t)-plane, including a(x) ≤ tb(x), x0xx1. Also suppose that the functions a(x) and b(x) are both continuous and both have continuous derivatives for x0xx1. Then for x0xx1:

This formula is the general form of the Leibniz integral rule and can be derived using the fundamental theorem of calculus. The (first) fundamental theorem of calculus is just a particular case of the above formula, for a(x) = a, a constant, b(x) = x and f(x, t) = f(t).

If both upper and lower limits are taken as constants, then the formula takes the shape of an operator equation:

ItDx = DxIt,

where Dx is the partial derivative with respect to x and It is the integral operator with respect to t 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:

  • the interchange of a derivative and an integral (differentiation under the integral sign; i.e., Leibniz integral rule)
  • the change of order of partial derivatives
  • the change of order of integration (integration under the integral sign; i.e., Fubini's theorem)

Higher dimensions[edit]

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 dynamics as the Reynolds transport theorem:

where is a scalar function, D(t) and ∂D(t) denote a time-varying connected region of R3 and its boundary, respectively, is the Eulerian velocity of the boundary (see Lagrangian and Eulerian coordinates) and dΣ = n dS is the unit normal component of the surface element.

The general statement of the Leibniz integral rule requires concepts from differential geometry, specifically differential forms, exterior derivatives, wedge products and interior products. With those tools, the Leibniz integral rule in p-dimensions is:[1]

where Ω(t) is a time-varying domain of integration, ω is a p-form, is the vector field of the velocity, , i denotes the interior product, dxω is the exterior derivative of ω with respect to the space variables only and is the time-derivative of ω.

Proof of Theorem[edit]

Lemma. One has:

Proof. From proof of the fundamental theorem of calculus,

Suppose a and b are constant, and that f(x) involves a parameter α which is constant in the integration but may vary to form different integrals. Assuming that f(x, α) is a continuous function of x and α in the compact set {(x, α) : α0 ≤ α ≤ α1 and axb}, and that the partial derivative fα(x, α) exists and is continuous, then if one defines:

may be differentiated with respect to α by differentiating under the integral sign; i.e.,

By the Heine–Cantor theorem it is uniformly continuous in that set. In other words for any ε > 0 there exists Δα such that for all values of x in [a, b]:

On the other hand:

Hence φ(α) is a continuous function.

Similarly if exists and is continuous, then for all ε > 0 there exists Δα such that:



Now, ε → 0 as Δα → 0, therefore,

This is the formula we set out to prove.

Now, suppose

where a and b are functions of α which take increments Δa and Δb, respectively, when α is increased by Δα. Then,

A form of the mean value theorem, where a < ξ < b, can be applied to the first and last integrals of the formula for Δφ above, resulting in

Dividing by Δα, letting Δα → 0, noticing ξ1a and ξ2b and using the above derivation for


This is the general form of the Leibniz integral rule.


General examples[edit]

Example 1[edit]

The function under the integral sign is not continuous at the point (x, α) = (0, 0) and the function φ(α) has a discontinuity at α = 0, because φ(α) approaches ±π/2 as α → 0±.

If we now differentiate φ(α) with respect to α under the integral sign, we get

which is, of course, true for all values of α except α = 0.

Example 2[edit]

An example with variable limits:

Examples for evaluating a definite integral[edit]

Example 3[edit]

The principle of differentiating under the integral sign may sometimes be used to evaluate a definite integral. Consider:


Now as x varies from 0 to π we have:



Integrating both sides with respect to α, we get:

C1 = 0 follows from evaluating φ(0):

To determine C2 in the same manner, we should need to substitute in a value of α greater than 1 in φ(α). This is somewhat inconvenient. Instead, we substitute α = 1/β, where |β| < 1. Then,

Therefore, C2 = 0.

The definition of φ(α) is now complete:

The foregoing discussion, of course, does not apply when α = ±1, since the conditions for differentiability are not met.

Example 4[edit]

First we calculate:

The limits of integration being independent of a, But we have:

On the other hand:

Equating these two relations then yields

In a similar fashion, pursuing yields

Adding the two results then produces

Note that if we define

it can easily be shown that

Given I1, this partial-derivative-based recursive relation (i.e., integral reduction formula) can then be utilized to compute all of the values of In for n > 1.

Example 5[edit]

Here, we consider the integral

Differentiating under the integral with respect to α, we have


However by definition, I(π/2) = 0, hence: C = π2/8 and

Example 6[edit]

Here, we consider the integral

We introduce a new variable φ and rewrite the integral as

Note that for φ = 1 we recover the original integral, now we proceed:

Integrating both sides of with respect to φ between the limits 0 and 1 yields

Therefore f(1) = f(0) however we note that from the equation for f(φ), we have f(0) = 2π, therefore the value of f at φ = 1, which is the same as the integral we set out to compute is 2π.

Other problems to solve[edit]

There are innumerable other integrals that can be solved "quickly" using the technique of differentiation under the integral sign. For example consider the following cases where one adds a new variable α:

The first integral, the Dirichlet integral, is absolutely convergent for positive α but only conditionally convergent when α is 0. Therefore differentiation under the integral sign is easy to justify when α > 0, but proving that the resulting formula remains valid when α is 0 requires some careful work.

Applications to series[edit]

Differentiating under the integral can also be applied to differentiating under summation, interpreting summation as counting measure. An example of an application is the fact that power series are differentiable in their radius of convergence.

Popular culture[edit]

Differentiation under the integral sign is mentioned in the late physicist Richard Feynman's best-selling memoir Surely You're Joking, Mr. Feynman! (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 Technology) while in high school. The technique was not often taught when Feynman later received his formal education in calculus 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 University. The direct quotation from Surely You're Joking, Mr. Feynman! regarding the method of differentiation under the integral sign is as follows:

One thing I never did learn was contour integration. I had learned to do integrals by various methods shown in a book that my high school physics teacher Mr. Bader had given me. One day he told me to stay after class. "Feynman," he said, "you talk too much and you make too much noise. I know why. You're bored. So I'm going to give you a book. You go up there in the back, in the corner, and study this book, and when you know everything that's in this book, you can talk again." So every physics class, I paid no attention to what was going on with Pascal's Law, or whatever they were doing. I was up in the back with this book: "Advanced Calculus", by Woods. Bader knew I had studied "Calculus for the Practical Man" a little bit, so he gave me the real works—it was for a junior or senior course in college. It had Fourier series, Bessel functions, determinants, elliptic functions—all kinds of wonderful stuff that I didn't know anything about. That book also showed how to differentiate parameters under the integral sign—it's a certain operation. It turns out that's not taught very much in the universities; they don't emphasize it. But I caught on how to use that method, and I used that one damn tool again and again. So because I was self-taught using that book, I had peculiar methods of doing integrals. The result was, when guys at MIT or Princeton had trouble doing a certain integral, it was because they couldn't do it with the standard methods they had learned in school. If it was contour integration, they would have found it; if it was a simple series expansion, they would have found it. Then I come along and try differentiating under the integral sign, and often it worked. So I got a great reputation for doing integrals, only because my box of tools was different from everybody else's, and they had tried all their tools on it before giving the problem to me.

See also[edit]


  • Frederick S. Woods (1934). Advanced Calculus (New ed.). Ginn and Company. ASIN B0006AMNBI. 
  • Frederick S. Woods (1926). Advanced Calculus (1st ed.). Ginn and Company. ASIN B00085L67S. 
  • David V. Widder (Jul 1990). Advanced Calculus (New ed.). Dover Publications Inc. ISBN 978-0-486-66103-2. 

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