Alphabeta Math

Improper Integrals: Examples and Counterexamples

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✓ 12 results · all verified · 0 also independently AI-judged
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1 · Prerequisites
Real Analysis
Set Theory
Topology
2 · Logical flowchart

3 · Items

Example

1/x on [-1,1] has principal value 0 but no improper integral

Example

A step function whose improper integral is the alternating harmonic series

Example

A positive continuous integrand can have finite integral while unbounded on every tail

Example

∫₀¹ x^-1/2 dx=2

Example

The rational p-threshold reverses between zero and infinity

Example

A rational-kernel Frullani integral

Example

∫_-∞^∞(1+x²)⁻¹ dx converges absolutely

Example

The substitution x=1/t exchanges the two rational p-tests

Counterexample

A bounded truncation function need not have an improper limit

Example

|x-c|^-1/2 has a convergent improper integral across an interior singularity

Counterexample

1/x² has no finite Cauchy principal value at zero

Example

Convergence range of x⁻ᵖ(1+x)^-q on (0,∞) for rational exponents

Sources