To calculate the indefinite integral, we'll use the substitution method.
We'll note y = f(x)
We'll calculate Integral of f(x) = y = x^3/(x^4+1).
We notice that if we'll differentiate x^4+1, we'll get 4x^3.
So, we'll note x^4+1 = t
(x^4+1)'dx = dt
(4x^3)dx = dt => (x^3)dx = dt/4
We'll re-write the integral in the variable t:
Int (x^3)dx/(x^4+1) = Int dt / 4t
Int dt / 4t= (1/4)*Int dt / t
(1/4)*Int dt / t = (1/4)*ln t + C
But x^4+1 = t.
Int (x^3)dx/(x^4+1) = (1/4)*ln(x^4+1) + C, where C is a family of constants.