We have f'(x) = 4x^3 + 2x

Now to find f(x) we have to integrate f'(x) = 4x^3 + 2x

Int [f'(x)] = f(x) = Int[ 4x^3 + 2x]

=> Int 4x^3 + Int 2x

=> 4x^4 / 4 + 2x^2 / 2 +C

=> x^4 + x^2 +C

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

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

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

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

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

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

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

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

According to the rule, f(x) could be determined evaluating the indefinite integral of f'(x)

Int (4x^3+2x)dx

We'll apply the additive property of integrals:

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

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

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

We'll simplify and we'll get:

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

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