f'(x) = 4x^3+2x.

Therefore the f(x) is an indefinite integral of the 4x^3+2x.

So f(x) = Integral {4x^3+2x}dx

f(x) = Integral 4x^3 dx +Integral (2x) dx constant

f(x) = 4*Integral x^3 dx + 2 Integral x dx + C

f(x) = 4 (1/(3+1)) {x^(3+1)} + 2 (1/(1+1))(x^(1+1) } +C

f(x) = x^4+ x^2 + C

To calculate a function, when knowing it's derivative, we'll have to integrate the expression of derivative.

We'll determine the indefinite integral of f'(x)= 4x^3 + 2x.

Int f'(x)dx = f(x) + C

Int (4x^3 + 2x)dx

We'll use the property of the indefinite integral, to be additive:

Int (4x^3 + 2x)dx =Int (4x^3)dx + Int (2x)dx

Int (4x^3)dx = 4*x^(3+1)/(3+1) + C

Int (4x^3)dx = 4x^4/4 + C

Int (4x^3)dx = x^4 + C (1)

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

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

We'll add: (1)+(2)

Int (4x^3 + 2x)dx = x^4 + x^2 + C

So, the function is:

** f(x) = x^4 + x^2**

We are given f'(x)= 4x^3 + 2x. Now f(x) is the indefinite integral of f'(x).

Now we use the relation that the integral of a*x^n is a/(n+1)*x^(n+1)

The integral of 4x^3 + 2x is

4 * x^(3+1) / (3+1) + 2* x^2 /2

= (4 / 4) * x^4 + (2 / 2)* x^2

= x^4 + x^2

**The function f(x) is x^4 + x^2**