Math 2.0

Real functions Proposition

Integral of the exponential function

Builds on Exponential functionThe exponential function is its own derivativeSecond fundamental theorem of calculus

Statement

For all real numbers aa and bb, ∫abex dx=eb−ea.\int_a^b e^x \, dx = e^b - e^a.

Remarks

The exponential function is its own derivative, so it is its own antiderivative. More generally, ∫abecx dx=(ecb−eca)/c\int_a^b e^{c x} \, dx = (e^{c b} - e^{c a})/c for c≠0c \ne 0. Mathlib states it as integral_exp.

Reading path

To understand this page from the ground up, read these 50 pages in order. The foundations come first.

  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. Geometric sequence definition
  16. Limit of a sequence definition
  17. Monotone sequence definition
  18. Series definition
  19. Continuous function definition
  20. Fermat's theorem on stationary points statement
  21. Sum of a finite geometric series statement
  22. Monotone convergence theorem for sequences statement
  23. Convergent sequence definition
  24. Squeeze theorem statement
  25. Every sequence has a monotone subsequence statement
  26. Cauchy product definition
  27. Partial sum definition
  28. Power series definition
  29. Bolzano–Weierstrass theorem statement
  30. The sequence r^n tends to zero for |r| < 1 example
  31. Radius of convergence definition
  32. Convergent series definition
  33. Boundedness theorem statement
  34. A series of nonnegative terms converges iff its partial sums are bounded statement
  35. Absolute convergence definition
  36. Sum of a series definition
  37. Extreme value theorem statement
  38. Comparison test statement
  39. Geometric series example
  40. Cauchy product of absolutely convergent series statement
  41. Rolle's theorem statement
  42. Ratio test statement
  43. Mean value theorem statement
  44. Ratio formula for the radius of convergence statement
  45. The exponential series has infinite radius of convergence statement
  46. Second fundamental theorem of calculus statement
  47. Exponential function definition
  48. Exponential of a sum statement
  49. The exponential function is its own derivative statement
  50. Integral of the exponential function statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Lists_of_integrals, Exponential functions
About this page
Tag06D0EWR2DV
URIm2://06D0EWR2DV
Kindstatement, proposition
WritingW0 stub
MSC26A36
StatusThis copy of the site has no build data. Run m2 status 06D0EWR2DV.
UsesExponential function, The exponential function is its own derivative, Second fundamental theorem of calculus

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by