Math 2.0

statement theorem

Integration by substitution

Statement

Let g ⁣:[a,b]→Rg \colon [a, b] \to \mathbb{R} be continuously differentiable, and let ff be continuous on an interval that contains g([a,b])g([a, b]). Then ∫abf(g(x)) g′(x) dx=∫g(a)g(b)f(u) du.\int_a^b f(g(x)) \, g'(x) \, dx = \int_{g(a)}^{g(b)} f(u) \, du.

Remarks

Let FF be an antiderivative of ff. By the chain rule, F∘gF \circ g is an antiderivative of (f∘g) g′(f \circ g) \, g'. Both sides equal F(g(b))−F(g(a))F(g(b)) - F(g(a)). The function gg need not be injective. For example, with u=x2u = x^2, ∫012xcos⁡(x2) dx=∫01cos⁡u du=sin⁡1\int_0^1 2x \cos(x^2) \, dx = \int_0^1 \cos u \, du = \sin 1. Mathlib states it as intervalIntegral.integral_comp_mul_deriv.

Explanations

L1 human

Substitution is the chain rule read backwards. We replace an expression g(x)g(x) by a new variable uu, and g′(x) dxg'(x) \, dx by dudu.

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  • edge proves in: at least one proof node (0 of 1)
  • edge instance_of or illustrates in: one example that shows the statement at work (0 of 1)
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What this needs

To read this node, read these 52 nodes in this order. The foundations come first, and the list ends with this node.

  1. Limit point of a set definition
  2. Local maximum definition
  3. Partition of an interval definition
  4. Sequence definition
  5. Limit of a function definition
  6. Local minimum definition
  7. Mesh of a partition definition
  8. Riemann sum definition
  9. Real sequence definition
  10. Subsequence definition
  11. Continuity at a point definition
  12. Derivative definition
  13. Riemann integrable function definition
  14. Bounded sequence definition
  15. Limit of a sequence definition
  16. Monotone sequence definition
  17. Continuous function definition
  18. Epsilon-delta definition of continuity statement
  19. Antiderivative definition
  20. Differentiable function definition
  21. Fermat's theorem on stationary points statement
  22. Riemann integral definition
  23. Upper and lower Darboux sums definition
  24. Convergent sequence definition
  25. Monotone convergence theorem for sequences statement
  26. Every sequence has a monotone subsequence statement
  27. Intermediate value theorem statement
  28. Composition of continuous functions is continuous statement
  29. Uniform continuity definition
  30. Derivative function definition
  31. Definite integral definition
  32. Upper and lower Darboux integrals definition
  33. Bolzano–Weierstrass theorem statement
  34. Chain rule statement
  35. Uniform continuity on a set definition
  36. Boundedness theorem statement
  37. Higher derivative definition
  38. Monotonicity of the integral statement
  39. Additivity of the integral over intervals statement
  40. Darboux and Riemann integrals agree statement
  41. Extreme value theorem statement
  42. Continuously differentiable function definition
  43. Continuous functions on a closed bounded interval are uniformly continuous statement
  44. Riemann's criterion for integrability statement
  45. Mean value theorem for integrals statement
  46. Rolle's theorem statement
  47. Continuous functions are Riemann integrable statement
  48. Mean value theorem statement
  49. First fundamental theorem of calculus statement
  50. Second fundamental theorem of calculus statement
  51. The integral of a continuous function is an antiderivative statement
  52. Integration by substitution statement

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Sources

  1. Web https://en.wikipedia.org/wiki/Integration_by_substitution, Statement for definite integrals

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