Math 2.0

concept

Reduction formula

Summary

A reduction formula expresses an integral that depends on an integer nn through the same integral with a smaller nn. Repeated use of the formula reduces the integral to a simple case. Integration by parts usually produces the formula.

Remarks

For In=∫0π/2sin⁡nx dxI_n = \int_0^{\pi/2} \sin^n x \, dx, integration by parts gives In=n−1nIn−2I_n = \frac{n - 1}{n} I_{n-2} for n≥2n \ge 2. With I0=π/2I_0 = \pi/2 and I1=1I_1 = 1, this gives all InI_n. These integrals lead to the Wallis product.

Explanations

L1 human

A reduction formula is a recursion for a family of integrals. Each step lowers a power, until the integral is easy.

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To read this node, read these 35 nodes in this order. The foundations come first, and the list ends with this node.

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 levelDerivativedefinitionRiemann 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…statementConvergent sequence, definition, no evidence levelConvergent sequencedefinitionMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementEvery sequence has a monotone subsequence, statement, no evidence levelEvery sequence has a …statementDifferentiable functions are continuous, statement, no evidence levelDifferentiable functi…statementDerivative function, definition, no evidence levelDerivative functiondefinitionBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementProduct rule, statement, no evidence levelProduct rulestatementBoundedness theorem, statement, no evidence levelBoundedness theoremstatementHigher derivative, definition, no evidence levelHigher derivativedefinitionExtreme value theorem, statement, no evidence levelExtreme value theoremstatementContinuously differentiable function, definition, no evidence levelContinuously differen…definitionRolle'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 partsstatementReduction formula, concept, no evidence levelReduction formulaconcept
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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. Riemann integrable function definition
  14. Bounded sequence definition
  15. Limit of a sequence definition
  16. Monotone sequence definition
  17. Continuous function definition
  18. Differentiable function definition
  19. Fermat's theorem on stationary points statement
  20. Convergent sequence definition
  21. Monotone convergence theorem for sequences statement
  22. Every sequence has a monotone subsequence statement
  23. Differentiable functions are continuous statement
  24. Derivative function definition
  25. Bolzano–Weierstrass theorem statement
  26. Product rule statement
  27. Boundedness theorem statement
  28. Higher derivative definition
  29. Extreme value theorem statement
  30. Continuously differentiable function definition
  31. Rolle's theorem statement
  32. Mean value theorem statement
  33. Second fundamental theorem of calculus statement
  34. Integration by parts statement
  35. Reduction formula concept

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Sources

  1. Web https://en.wikipedia.org/wiki/Integration_by_reduction_formulae, How to find the reduction formula

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