### What is the primitive of a function and how do you determine the primitive of a function?

Let \left[a,b\right] be an interval, and f : \left[a,b\right] \to \mathbb{R} a function defined on the interval \left[a,b\right]. We say that f admits a primitive on \left[a,b\right] if there exists a differentiable function F : \left[a,b\right]\to \mathbb{R} such that for any x \in \left]a,b\right[, F^{\prime}(x) = f(x).
We then say that F is a primitive of f.

For example, for f(x) = 3 x^2 + 5, a primitive of f on \mathbb{R} is F(x) = x^3 + 5 x. This can be verified by deriving F.
We then say that F is a primitive of f.

The following table shows the primitives of some common functions.
We then say that F is a primitive of f.

Cliquez sur le tableau pour l'afficher en plein écran.

### Sufficient condition for the existence of a primitive

Let \left[a,b\right] be an interval, and f : \left[a,b\right] \to \mathbb{R} a function defined on the interval \left[a,b\right].
We then say that F is a primitive of f.

In this case, F is the only primitive of f that cancels at a. This result is known as the fundamental theorem of analysis.

So, if a function is continuous over an interval, it admits a primitive over the interval.

### Integral and primitive relationship

Knowing a primitive of a function f allows you to calculate its integral over segments.

Indeed, if f is a continuous function defined on \left[a,b\right] and if F is a primitive of f, then we have

\int_{a}^{b} f(x)dx = {\left[F(x)\right]}_a^b = F(b) - F(a)

\int_{a}^{b} f(x)dx = {\left[F(x)\right]}_a^b = F(b) - F(a)

### Properties on primitives

It is possible to state a number of relationships which follow from the derivation formulae. Here we consider two derivable functions f and g defined on an interval I. The table below summarises the primitives of the main operations on functions.

Cliquez sur le tableau pour l'afficher en plein écran.

You now know what a primitive is and how do you determine the primitive of a function. Primitives are mainly involved in the calculation of integrals, and are closely related to the derivation of functions.

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