Math 2.0

example

Dirichlet integral

Example

The improper integral of sin⁡xx\frac{\sin x}{x} over (0,∞)(0, \infty) converges, and ∫0∞sin⁡xx dx=π2.\int_0^\infty \frac{\sin x}{x} \, dx = \frac{\pi}{2}. The integrand extends continuously to x=0x = 0 with value 11. So only the upper limit is improper. Integration by parts gives ∫1bsin⁡xx dx=cos⁡1−cos⁡bb−∫1bcos⁡xx2 dx\int_1^b \frac{\sin x}{x} \, dx = \cos 1 - \frac{\cos b}{b} - \int_1^b \frac{\cos x}{x^2} \, dx. The last integral converges absolutely, so the limit exists.

Remarks

The integral does not converge absolutely: ∫0∞∣sin⁡x∣x dx=∞\int_0^\infty \frac{|\sin x|}{x} \, dx = \infty. So sin⁡xx\frac{\sin x}{x} is not Lebesgue integrable on (0,∞)(0, \infty). The value π/2\pi/2 needs other methods, such as differentiation under the integral sign or contour integration.

Explanations

L1 human

The area under sin⁡xx\frac{\sin x}{x} alternates in sign, and the pieces shrink. The positive and negative pieces cancel to give the value π/2\pi/2.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionLocal maximum, definition, no evidence levelLocal maximumdefinitionPartition of an interval, definition, no evidence levelPartition of an inter…definitionSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionLocal minimum, definition, no evidence levelLocal minimumdefinitionMesh of a partition, definition, no evidence levelMesh of a partitiondefinitionRiemann sum, definition, no evidence levelRiemann sumdefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSubsequence, definition, no evidence levelSubsequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionDerivative, definition, no evidence levelDerivativedefinitionLimit at infinity, definition, no evidence levelLimit at infinitydefinitionOne-sided limit, definition, no evidence levelOne-sided limitdefinitionRiemann integrable function, definition, no evidence levelRiemann integrable fu…definitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionDifferentiable function, definition, no evidence levelDifferentiable functi…definitionFermat's theorem on stationary points, statement, no evidence levelFermat's theorem on s…statementRiemann integral, definition, no evidence levelRiemann integraldefinitionConvergent sequence, definition, no evidence levelConvergent sequencedefinitionEvery sequence has a monotone subsequence, statement, no evidence levelEvery sequence has a …statementMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementDifferentiable functions are continuous, statement, no evidence levelDifferentiable functi…statementDerivative function, definition, no evidence levelDerivative functiondefinitionImproper integral, definition, no evidence levelImproper integraldefinitionBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementProduct rule, statement, no evidence levelProduct rulestatementHigher derivative, definition, no evidence levelHigher derivativedefinitionBoundedness theorem, statement, no evidence levelBoundedness theoremstatementAbsolutely convergent improper integral, definition, no evidence levelAbsolutely convergent…definitionContinuously differentiable function, definition, no evidence levelContinuously differen…definitionExtreme value theorem, statement, no evidence levelExtreme value theoremstatementRolle's theorem, statement, no evidence levelRolle's theoremstatementMean value theorem, statement, no evidence levelMean value theoremstatementSecond fundamental theorem of calculus, statement, no evidence levelSecond fundamental th…statementIntegration by parts, statement, no evidence levelIntegration by partsstatementDirichlet integral, example, no evidence levelDirichlet integralexample
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  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. Limit at infinity definition
  14. One-sided limit definition
  15. Riemann integrable function definition
  16. Bounded sequence definition
  17. Limit of a sequence definition
  18. Monotone sequence definition
  19. Continuous function definition
  20. Differentiable function definition
  21. Fermat's theorem on stationary points statement
  22. Riemann integral definition
  23. Convergent sequence definition
  24. Every sequence has a monotone subsequence statement
  25. Monotone convergence theorem for sequences statement
  26. Differentiable functions are continuous statement
  27. Derivative function definition
  28. Improper integral definition
  29. Bolzano–Weierstrass theorem statement
  30. Product rule statement
  31. Higher derivative definition
  32. Boundedness theorem statement
  33. Absolutely convergent improper integral definition
  34. Continuously differentiable function definition
  35. Extreme value theorem statement
  36. Rolle's theorem statement
  37. Mean value theorem statement
  38. Second fundamental theorem of calculus statement
  39. Integration by parts statement
  40. Dirichlet integral example

Sources

  1. Web https://en.wikipedia.org/wiki/Dirichlet_integral, lead section
  2. Web https://en.wikipedia.org/wiki/Improper_integral, Improper Riemann integrals and Lebesgue integrals

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