Alphabeta Math

Properties of the Integral and the Working FTC: Examples and Counterexamples

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✓ 12 results · all verified · 7 also independently AI-judged
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1 · Prerequisites
Abstract Algebra
Number Theory
Real Analysis
Set Theory
Topology
2 · Logical flowchart

3 · Items

Example

∫₀¹ xᵐ = 1/ι(m+1), computed by the fundamental theorem and checked against the definition

Example

A step function integrated by additivity over subintervals, and the same value from the definition

Example

The integral test applied to ∑ 1/ι(k+1)ᵖ for rational p>0, cross-checked against the published p-series theorem

Example

H(x) = 2√x on [0,1]: H is continuous, H' is unbounded on (0,1], and H' is therefore not Riemann integrable

Counterexample

The sign function is Riemann integrable on [-1,1] and has no primitive there

Counterexample

A function differentiable on [0,1] whose derivative is unbounded, hence not Riemann integrable

Counterexample

Integrable φ and integrable f with φ∘ f not integrable: the order of the hypotheses in the composition theorem cannot be reversed

Counterexample

A function that is not Riemann integrable although | f| is

Counterexample

Continuous fₙ → 0 pointwise on [0,1] with ∫₀¹ fₙ = 1 for every n

Counterexample

Continuous f and integrable sign-changing g with ∫ₐᵇ fg ≠ f(ξ)∫ₐᵇ g for every ξ

False statement

FALSE: if u and v are differentiable on [a,b] then ∫ₐᵇ uv' = u(b)v(b)-u(a)v(a)-∫ₐᵇ u'v

False statement

FALSE: in the substitution theorem the continuity of f may be weakened to integrability, f∘φ still being integrable

False statement

FALSE: for every integrable f on [a,b], the integral function F(x)=∫ₐˣ f satisfies F' = f on [a,b]

Sources