Math 2.0

statement proposition

Integral of the exponential function

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.

Explanations

L1 human

The exponential function does not change under differentiation. So it also does not change under integration, up to the constant.

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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. 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
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  27. Partial sum definition
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  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

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Sources

  1. Web https://en.wikipedia.org/wiki/Lists_of_integrals, Exponential functions

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