The concept of a primitive

What is an antiderivative?

An antiderivative of a function f on an interval I is a function F that is differentiable on I such that F'(x) = f(x) for all x in I. Finding an antiderivative therefore amounts to performing the inverse operation of differentiation.

Example: if f(x) = 2x, then F(x) = x² is an antiderivative of f, since F'(x) = 2x = f(x).

An infinite number of antiderivatives

If F is an antiderivative of f, then any function of the form F(x) + k, where k is a real constant, is also an antiderivative of f: indeed, the derivative of a constant is zero. We say that the antiderivatives of f form a family F(x) + k, where k belongs to R.

Common antiderivatives

Function f(x)	Antiderivative F(x)
x^n (n ≠ -1)	x^(n+1)/(n+1) + k
1/x (x > 0)	ln(x) + k
e^x	e^x + k
cos(x)	sin(x) + k
sin(x)	-cos(x) + k

Determining the constant

To find THE antiderivative that satisfies a specific condition, we use a given value. Example: find the antiderivative F of f(x) = 2x such that F(0) = 3. We have F(x) = x² + k, so F(0) = 0 + k = 3, hence k = 3 and F(x) = x² + 3.

Common pitfall

Never forget the constant k when searching for an antiderivative in general. Simply writing F(x) = x² as the answer to the question “What are the antiderivatives of 2x?” is incomplete: one must write F(x) = x² + k, where k belongs to R.