List of basic formulas related to the definite integral:
- 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) ). - 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
]
- Integral of a constant:
[
\int_{a}^{b} c \, dx = c(b – a)
] - 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
] - Integrate from the exponent:
[
\int_{a}^{b} e^{x} \, dx = e^{b} – e^{a}
] - 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)
]