Some Properties of Definite Integrals#
A definite integral, denoted by the symbol (\int_a^b f(x) dx), represents the area under the curve of the function (f(x)) between the bounds (a) and (b).
Here are some of the properties of definite integrals:
- Linearity:
∫ab(c1f(x)+c2g(x))dx=c1∫abf(x)dx+c2∫abg(x)dx
where (c_1) and (c_2) are constants.
- Constant Factor Rule:
∫abcf(x)dx=c∫abf(x)dx
- Sum Rule:
∫ab(f(x)+g(x))dx=∫abf(x)dx+∫abg(x)dx
- Chain Rule:
∫abf(g(x))g′(x)dx=∫g(a)g(b)f(u)du
- Indefinite Integral:
∫abf(x)dx=F(b)−F(a) where (F(x)) is an antiderivative of (f(x)).
- Value of a definite integral:
∫abf(x)dx=F(b)−F(a), where (F(x)) is an antiderivative of (f(x)).
These properties allow us to manipulate definite integrals in a systematic way, making it easier to evaluate them