**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.

Noah Dana-Picard
2007-12-28