# Primitive. Indefinite integral

*Primitive. Finding of primitive: infinite set of solutions.*

Indefinite integral. Constant of integration.

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* ) = 3*x *^{ 2 } on the

interval ( – , +

)

, because

*F’*(

*x*)

*=*

(

*x*

^{ 3}

*)*

*’*= 3

*x*

^{2}

=

*f*(

*x*)

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

x

on a segment

*X*is a

set of

set of

*all*

its

its

*primitives*.

This is written as :

where *C – *any constant, called a *constant of integration*.