Solve the indefinite integral of y=x^3/(x^4+1)

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.