The area under the curve y= x+ 6/x and x axis is calculated using Leibniz-Newton formula.

Int ( x+ 6/x)dx = F(b) - F(a), where a = 1 and b = 2

First, we'll determine the result of the indefinite integral:

Int ( x+ 6/x)dx

We'll use the additive property of integrals:

Int ( x+ 6/x)dx = Int xdx + Int 6dx/x

We'll simplify and we'll take out the constants and we'll get:

Int ( x+ 6/x)dx = Int xdx + 6Int dx/x

Int ( x+ 6/x)dx = x^2/2 + 6ln |x| + C

The resulted expression is F(x).

Now, we'll determine F(b) = F(2):

F(2) = 6ln |2| + 2^2/2

F(2) = 6ln |2| + 2

Now, we'll determine F(a) = F(1):

F(1) = 6ln |1| + 1^2/2

F(1) = 0 + 1/2

We'll determine the area:

A = F(2) - F(1)

A = 6ln |2| + 2 - 1/2

**A = ln 2^6 + 3/2**

**A = ln 64 + 1.5 square units**

**A = 4.15 + 1.5**

**A = 5.65 square units**