find f(x) if f'(x) = 3x^2 - 5x + 3

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 (3x^2 - 5x +
3)dx

We'll apply the additive property of indefinite
integrals:

Int (3x^2 - 5x + 3)dx = Int 3x^2dx - Int 5xdx +
Int 3dx

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

We'll simplify and we'll
get:

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

Int 5xdx = 5x^2/2 + C
(2)

Int 3dx = 3x + C (3)

We'll
add  (1),(2),(3):

f(x)
=  (1)+(2)+(3)

f(x) = x^3 + 5x^2/2 + 3x +
C

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