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