Determine the area of the surface between the lines x=0,x=1 and the curve f(x)=(3x^2 + 3)/(x^3 + 3x).

The area which has to be determined is bounded by the
given curve f(x), the lines x = 0 and x = 1, also the x
axis.

To calculate the area, we'll use the
formula:

S = Integral (f(x) - ox)dx=Int f(x)dx = Int(3x^2 +
3)dx/(x^3 + 3x).

We'll calculate the integral, using
substitution technique.

We'll note x^3 + 3x =
t.

We'll differentiate x^3 +
3x.

(x^3 + 3x)' =
t'

(3x^2+3)dx = dt

We notice
that the result of differentiating the function is the numerator of the
function.

We'll re-write the
integral:

Int dt/t = ln t = ln (x^3 + 3x) +
C

Now, we'll calculate the value of the area, using
Leibnitz Newton formula::

S = F(1) - F(0),
where

F(1) = ln (1^3 + 3*1) = ln
4

F(0) = ln (0^3 + 3*0) = ln 0
impossible!

S = ln 4 - infinite,
impossible!