What is f(x) if f'(x)=6x^2-2x+1 ?

We know that F'(x) = f(x), where Int f'(x) dx =
f(x).

We'll apply the indefinite integral to the expression
of f(x):

Int f'(x) dx = Int
(6x^2-2x+1)dx

We'll apply the additive property of
indefinite integrals:

Int (6x^2-2x+1)dx = Int 6x^2dx - Int
2xdx + Int dx

Int 6x^2dx = 6x^3/3 +
C

We'll simplify and we'll
get:

Int 6x^2dx = 2x^3 + C
(1)

Int 2xdx = 2x^2/2 +
C

We'll simplify and we'll
get:

Int 2xdx = x^2 + C
(2)

Int dx = x + C (3)

We'll
add  (1),(2),(3):

f(x)
=  (1)+(2)+(3)

f(x) = 2x^3 + x^2 + x +
C

Note: C+C+C = C (family of
constants)