Integrals

Integration is the reciprocal process to differentiation, that is to say, given a function, f(x), determine the function, F(x) that after differentiating, is f(x).

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 or antiderivative, it has infinite antiderivatives differentiating them all into a constant.

[F(x) + C]' = F'(x) + 0 = F'(x) = f(x)

Indefinite Integral

The indefinite integral is the set of the infinite antiderivatives that a function can have.

It is denoted 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 is equal to 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

Indefinite Integral Formulas

Integral of X

Integral of a Constant

Integral of an Exponent

Integral

Exponential Integral

Exponential Integral

Integral of Sine

Integral of Cosine

Integral of Tangent

Integral of Cotangent

Integral of Arcsin

Integral of Arctangent