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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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