Primitive. Indefinite integral
Indefinite integral. Constant of integration.
Primitive. A continuous function F ( x ) is called a primitive for a function f ( x ) on a segment X , if for each
E x a m p l e . The function
F
(
x
) =
x
^{
3
}
is
a primitive for the function
f
(
x
) = 3
x
^{
2
}
on the
interval (

, +
)
,
because
for all
x
(

, +
)
.
It is easy to check, that the function
x
^{
3
}
+
13 has the same
derivative 3
x
^{
2
}
,
so it is also a primitive for the function
3
x
^{
2
}
for all
x
(

, +
) .
It is clear, that instead of 13 we can use any constant. Thus, the problem of finding a primitive has an infinite set of solutions. This fact is reflected in the definition of an indefinite integral:
Indefinite integral
of a function
f
(
x
)
on a segment
X
is a
set of
all
its
primitives
.
This is written as :
where C – any constant, called a constant of integration .