Math 2.0

Real functions Further topics Theorem

Cauchy formula for repeated integration

Builds on Continuous functionIterated integralFundamental theorem of calculus

Statement

Let a<ba < b be real numbers, let f ⁣:[a,b]→ℝf \colon [a, b] \to \R be continuous, and let n≥1n \ge 1 be an integer. Define F0=fF_0 = f and Fk(x)=∫axFk−1(t) dtF_k(x) = \int_a^x F_{k-1}(t) \, dt for k≥1k \ge 1 and x∈[a,b]x \in [a, b]. Then for all x∈[a,b]x \in [a, b], Fn(x)=1(n−1)!∫ax(x−t)n−1f(t) dt.F_n(x) = \frac{1}{(n - 1)!} \int_a^x (x - t)^{n - 1} f(t) \, dt.

Remarks

The proof is by induction on nn. The case n=1n = 1 is the definition. For the step, note that both sides vanish at x=ax = a. Differentiating the right side under the integral sign gives the same formula with n−1n - 1 in place of nn. The fundamental theorem of calculus then gives equality. Equivalently, one writes FnF_n as an iterated integral over a simplex and integrates out all variables but the last one. Example: for f=1f = 1, a=0a = 0 and n=2n = 2, the right side is ∫0x(x−t) dt=x2/2\int_0^x (x - t) \, dt = x^2 / 2, which is the twice repeated integral of 11.

Going deeper

For a student

The formula is used in the proof of Taylor's theorem with the integral remainder. Replacing (n−1)!(n - 1)! by Γ(n)\Gamma(n) and letting nn be any positive real number gives the Riemann–Liouville integral, the starting point of fractional calculus.

Reading path

To understand this page from the ground up, read these 42 pages in order. The foundations come first.

  1. Continuous function concept
  2. Euclidean space object
  3. Function concept
  4. Limit point of a set definition
  5. Local maximum definition
  6. Partition of an interval definition
  7. Sequence definition
  8. Function of several variables concept
  9. Limit of a function definition
  10. Local minimum definition
  11. Mesh of a partition definition
  12. Riemann sum definition
  13. Real sequence definition
  14. Subsequence definition
  15. Iterated integral concept
  16. Continuity at a point definition
  17. Derivative definition
  18. Riemann integrable function definition
  19. Bounded sequence definition
  20. Limit of a sequence definition
  21. Monotone sequence definition
  22. Continuous function definition
  23. Antiderivative definition
  24. Fermat's theorem on stationary points statement
  25. Riemann integral definition
  26. Convergent sequence definition
  27. Monotone convergence theorem for sequences statement
  28. Every sequence has a monotone subsequence statement
  29. Intermediate value theorem statement
  30. Definite integral definition
  31. Bolzano–Weierstrass theorem statement
  32. Additivity of the integral over intervals statement
  33. Boundedness theorem statement
  34. Monotonicity of the integral statement
  35. Extreme value theorem statement
  36. Mean value theorem for integrals statement
  37. Rolle's theorem statement
  38. First fundamental theorem of calculus statement
  39. Mean value theorem statement
  40. Second fundamental theorem of calculus statement
  41. Fundamental theorem of calculus statement
  42. Cauchy formula for repeated integration statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Cauchy_formula_for_repeated_integration
About this page
TagEYXG11V76E
URIm2://EYXG11V76E
Kindstatement, theorem
WritingW0 stub
MSC26A42, 26A33
StatusThis copy of the site has no build data. Run m2 status EYXG11V76E.
AboutIterated integral
UsesContinuous function, Iterated integral, Fundamental theorem of calculus
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Changed2 November 2026

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