Primitive. Indefinite integral
Primitive. Finding of primitive: infinite set of solutions.
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 
F’ ( x ) = f ( x ).
E x a m p l e . The function F ( x ) = x 3
is
a primitive for the function f ( x ) = 3x 2 on the
interval ( -  , +
)
, because
F’ ( x ) =
( x
3
)’ = 3x
2
=
f ( x )
for all x
 (
- , +
)
.
It is easy to check, that the function x
3
+
13 has the same derivative 3x
2,
so it is also a primitive for the function 3x 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.
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