Find the area under the curve y= x+ 6/x, x=1 to x=2.

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