Integrating is the reciprocal process to differentiating, that is, given a function, f(x), determine the function, F(x) that after differentiating, is f(x).
It is said then, that F(x) is an antiderivative or primitive of f(x), or that the antiderivatives of f(x) are differentiable functions, F(x), such that:
F'(x) = f(x).
If a function, f(x), has a primitive, it has infinite primitives differentiating them all into a constant.
[F(x) + C]' = F'(x) + 0 = F'(x) = f(x)
The indefinite integral is the set of the infinite primitives that a function can have.
It is represented by ∫f(x) dx.
∫ is the integral symbol.
f(x) is the integrand.
x is the integration variable.
C is the constant of integration.
dx is the differential of x, and indicates the variable of the function to be integrated.
If F(x) is an antiderivative of f(x), then:
∫f(x) dx = F(x) + C
Verify that the primitive function is accurate enough to derive.
Properties of the Indefinite Integral
1. The integral of a sum of functions equals the sum of the integrals of these functions.
∫[f(x) + g(x)] dx =∫ f(x) dx +∫ g(x) dx
2. The integral of the product of a constant for a function is equal to the constant for the integral of the function.
∫ k f(x) dx = k ∫f(x) dx