Theorems · Theorem · real analysis

intervalIntegral.integral_deriv_of_contDiffOn_Icc

∀ {E : Type u_3} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {f : ℝ → E} {a b : ℝ} [CompleteSpace E],
  ContDiffOn ℝ 1 f (Set.Icc a b) → a ≤ b → ∫ (x : ℝ) in a..b, deriv f x = f b - f a

Fundamental theorem of calculus-2: If f : ℝ → E is C^1 on [a, b], then ∫ y in a..b, deriv f y equals f b - f a.

Defined in

Mathlib.MeasureTheory.Integral.IntervalIntegral.ContDiff

Cited by

4 results in Mathlib

Foundations

Depth 267 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceCompleteSpace