List of basic formulas related to the definite integral:

  1. Definition of a definite integral:
    [
    \int_{a}^{b} f(x) \, dx = F(b) – F(a)
    ]
    where ( F(x) ) is the antiderivative of the function ( f(x) ).
  2. Properties of the definite integral:
  • Линейность:
    [
    \int_{a}^{b} [c \cdot f(x) + g(x)] \, dx = c \cdot \int_{a}^{b} f(x) \, dx + \int_{a}^{b} g(x) \, dx
    ]
  • The property of changing the limits of integration:
    [
    \int_{a}^{b} f(x) \, dx = -\int_{b}^{a} f(x) \, dx
    ]
  • Свойство аддитивности:
    [
    \int_{a}^{c} f(x) \, dx + \int_{c}^{b} f(x) \, dx = \int_{a}^{b} f(x) \, dx
    ]
  1. Integral of a constant:
    [
    \int_{a}^{b} c \, dx = c(b – a)
    ]
  2. The integral from the degree function is:
    [
    \int_{a}^{b} x^n \, dx = \frac{b^{n+1} – a^{n+1}}{n+1}, \quad n \neq −1
    ]
  3. Integrate from the exponent:
    [
    \int_{a}^{b} e^{x} \, dx = e^{b} – e^{a}
    ]
  4. Integral of trigonometric functions:
  • Для синуса:
    [
    \int_{a}^{b} \sin(x) \, dx = -\cos(b) + \cos(a)
    ]
  • For cosine:
    [
    \int_{a}^{b} \cos(x) \, dx = \sin(b) – \sin(a)
    ]

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