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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