Math 2.0

Real functions Proposition

Integral of a power

Builds on Power ruleSecond fundamental theorem of calculus

Statement

Let n≥0n \ge 0 be an integer and let a,b∈ℝa, b \in \mathbb{R}. Then ∫abxn dx=bn+1−an+1n+1.\int_a^b x^n \, dx = \frac{b^{n+1} - a^{n+1}}{n + 1}.

Remarks

By the power rule, xn+1/(n+1)x^{n+1}/(n + 1) is an antiderivative of xnx^n. Apply the second fundamental theorem. The same formula holds for each real exponent r≠−1r \ne -1, on intervals inside (0,∞)(0, \infty). The case r=−1r = -1 gives the logarithm. Mathlib states it as integral_pow.

Reading path

To understand this page from the ground up, read these 32 pages in order. The foundations come first.

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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 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…statementBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementBoundedness theorem, statement, no evidence levelBoundedness theoremstatementProduct rule, statement, no evidence levelProduct rulestatementExtreme value theorem, statement, no evidence levelExtreme value theoremstatementPower rule, statement, no evidence levelPower rulestatementRolle'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…statementIntegral of a power, statement, no evidence levelIntegral of a powerstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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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. Bolzano–Weierstrass theorem statement
  25. Boundedness theorem statement
  26. Product rule statement
  27. Extreme value theorem statement
  28. Power rule statement
  29. Rolle's theorem statement
  30. Mean value theorem statement
  31. Second fundamental theorem of calculus statement
  32. Integral of a power statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Lists_of_integrals, Integrals of simple functions
About this page
Tag0Q3HYFT9RQ
URIm2://0Q3HYFT9RQ
Kindstatement, proposition
WritingW0 stub
MSC26A36
StatusThis copy of the site has no build data. Run m2 status 0Q3HYFT9RQ.
UsesPower rule, Second fundamental theorem of calculus

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