Differentiating Under the Integral Sign
1. Introduction
The technique of differentiating under the integral signsometimes calledthe Leibniz rule or Feynmans trickallows a parameterderivative to be moved inside a definite integral. It is a powerful toolthat turns many seemingly intractable integrals into elementaryexpressions, and it can also be used to derive closedform results,evaluate series, and solve differential equations. The method was popularizedby physicist RichardFeynman, who famously used it to solve integrals thatstumped his professors.
2. The General Leibniz Rule
Suppose we have a function of two variables, \(f(x,t)\), that is integrablewith respect to \(x\) on an interval \([a,b]\) for every value of theparameter \(t\) in an open set \(I\). If the partial derivative\(\partial f/\partial t\) exists and is continuous on \([a,b]\times I\),then the derivative of the integral with respect to \(t\) can be takeninside the integral sign:
If the limits of integration also depend on \(t\)say they are\(a(t)\) and \(b(t)\)the rule expands to the full Leibniz formula:
3. When the Rule Applies Sufficient Conditions
The theorem is most often stated in the simple fixed limits form, butthe essential hypotheses are:
- Integrability: For each \(t\in I\), the function\(x\mapsto f(x,t)\) is integrable on \([a,b]\).
- Continuity of the partial derivative: The mixedpartial \(\partial f/\partial t\) exists and is continuous on\([a,b]\times I\). Continuity can be weakened to uniform integrability,but continuity is a convenient check for most textbook problems.
- Boundedness for interchange: There exists anintegrable function \(g(x)\) such that\(|\partial f/\partial t(x,t)|\le g(x)\) for all \(t\in I\). This ensuresthat the dominated convergence theorem justifies moving the limit inside.
If the limits of integration depend on \(t\), the same conditions must holdon the moving interval together with differentiability of the limits.
4. A Classic Example
Consider the integral\[I(\alpha)=\int_{0}^{\infty}\frac{e^{-\alpha x}-e^{-\beta x}}{x}\,dx ,\qquad \alpha,\beta>0 .\]Direct evaluation is difficult, but differentiating with respect to \(\alpha\)clarifies the structure.
5. Another Illustration Gaussian Integral
A celebrated application uses the trick to evaluate the Gaussian integral\(\displaystyle\int_{-\infty}^{\infty}e^{-x^{2}}dx\). Define\[F(a)=\int_{-\infty}^{\infty}e^{-ax^{2}}dx ,\qquad a>0 .\]Differentiating under the integral sign,\[F'(a)=\int_{-\infty}^{\infty}(-x^{2})e^{-ax^{2}}dx =-\,\frac{1}{2}\frac{d}{da}F(a) .\]Solving the firstorder differential equation \(F'(a)=-\frac12F(a)/a\) yields\(F(a)=C a^{-1/2}\). The constant \(C\) is determined by evaluating\(F(1)=\sqrt{\pi}\), giving the classic result\[\int_{-\infty}^{\infty}e^{-x^{2}}dx=\sqrt{\pi}.\]
6. Applications in Physics and Engineering
The technique appears in many branches of applied mathematics:
- Heat conduction: Solutions of the heat equation ofteninvolve integrals of the form \(\int_{0}^{\infty}e^{-k^{2}t}\sin(kx)/k\,dk\). Differentiating with respect to the time parameter \(t\) simplifiesthe integral, leading to closedform expressions for temperature profiles.
- Probability theory: Moment generating functions aredefined as integrals of exponential families. The derivative of the mgfwith respect to its parameter yields moments, and the interchange isjustified by the same conditions discussed above.
- Quantum mechanics: Pathintegral formulationsfrequently involve parameterdependent integrals; the Leibniz rule allowsthe extraction of propagator derivatives without cumbersome limitingprocesses.
7. Common Pitfalls and How to Avoid Them
Even seasoned practitioners can err when the hypotheses are ignored.
- Nonuniform convergence: If \(\partial f/\partial t\)fails to be dominated by an integrable function, interchanging limit andintegral may produce a wrong answer. Always verify a uniform bound oruse the dominated convergence theorem explicitly.
- Improper integrals at the endpoint: When the integrationrange is infinite, ensure that both the original integral and the integralof the partial derivative converge. Sometimes a parameter limit must betaken after the interchange.
- Discontinuous integrands: If \(f\) has jumpdiscontinuities that move with \(t\), the derivative of the integral acquiresadditional terms corresponding to the motion of the discontinuities. Thefull Leibniz rule with variable limits captures these contributions.
8. A StepbyStep Checklist
- Write the integral in the form \(\displaystyle I(t)=\int_{a}^{b} f(x,t)\,dx\).
- Check that \(f\) and \(\partial f/\partial t\) are continuous on thedomain; if not, find a dominating integrable function.
- Compute \(\displaystyle \frac{\partial}{\partial t}f(x,t)\).
- Integrate the resulting expression with respect to \(x\). This newintegral is often simpler or recognizable.
- If needed, solve the resulting differential equation for \(I(t)\) anddetermine any constants from known values (e.g., by setting a parameterto a convenient value).
9. Further Reading
For a deeper theoretical background see:
- G.B.Folland, Real Analysis, Chapter2 dominated convergence.
- R.P.Feynman and A.R.Hibbs, Quantum Mechanics and PathIntegrals, 2.4 the original Feynman trick.
- L.G.K.Thompson, Differentiation Under the Integral Sign, AmericanMathematical Monthly, 1975 a collection of classic examples.
10. Summary
Differentiating under the integral sign is a versatile method that turnsparameterdependent integrals into manageable expressions. The core of themethod is the Leibniz rule, whose validity rests on relatively modestregularity conditions. When applied correctly, it provides elegantsolutions to problems ranging from elementary calculus to advanced physics.Mastering the technique enriches a mathematicians toolkit and often yieldsinsight that would otherwise remain hidden.
After School Program Attendance Sign In/Sign Out Sheet and Reference File Download Link
Sample Sign Plans - Toronto Building - Sign By-law Unit and Reference File Download Link