# Integration

**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)**

## Indefinite Integral

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**