# Primitives - The undefinite integral.

Definition 8.1.1   Let and be two functions defined on the same interval . We say that is a primitive of on if is differentiable on and .

Example 8.1.2   The sine function is a primitive a the cosine function.

Theorem 8.1.3   Let and be two functions defined on the same interval . If is a primitive of on , then the functions , with constant , are all the primitives of .

This is a consequence of Lagrange's theorem (v.s. 8.5), via the following proposition:

Proposition 8.1.4   Let be a function, differentiable on the interval . If for any , , then is a constant function on .

We denote all the primitives of a given function by the undefinite integral:

You can compare the following theorem with Thm 4.4.

Theorem 8.1.5 (Table of usual primitives)

 conditions 0 1 , when

Noah Dana-Picard 2007-12-28