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
a primitive for the function f ( x ) = 3x 2 on the
F’ ( x ) =
)’ = 3x
f ( x )
interval ( - , +
for all x
- , +
It is easy to check, that the function x
13 has the same derivative 3x
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:
of a function f
on a segment X is a
set of all
This is written as :
where C – any constant, called a constant of integration.