Calculus of a Single Variable, Chapter 7, 7.4, Section 7.4, Problem 14

The arc length of a function is the length of the described curve within some interval on the x-axis (and a corresponding interval on the y-axis). If we were to lay a piece of string over the line of the graph in this window, the graph going from corner to diagonally opposite corner in a curve, the arc length is the length of the piece of string used.
In this case the function to find the arc length of is
y = ln ((e^x+1)/(e^x - 1))
where ln(x) is the natural logarithm of x (the inverse function to e^(x) , recall).
The window over which to find the length is the rectangle given by ln 2 <= x <= ln3 and ln2 <= y <= ln3.
The function for the arc length of a generic function y = f(x) is
s = int_a^b sqrt(1+((dy)/(dx))^2) \quad dx
This integrates along the line of the graph, adding infinitesimal sections together where each very small section added is a straight line diagonal over the small window x_0 <=x <=x_0 + dx , y_0 <=y <= y_0 + dy . Though y = f(x) may be a curve, the point is that these windows over which the length is integrated (added up) are so small that the graph is a straight line within them making it a simple thing to add the small (straight line) sections together. This concept is the basis of calculus.
In this example, since y = ln ((e^x+1)/(e^x-1)) = ln(e^x + 1) - ln(e^x-1) then its derivative (dy)/(dx) is given by
(dy)/(dx) = (1/(e^x+1))e^x - (1/(e^x-1))e^x = e^x(1/(e^x+1) - 1/(e^x-1)) = e^x((e^x-1-e^x-1)/((e^x+1)(e^x-1))) = -(2e^x)/((e^x+1)(e^x-1))
Now working in steps to find the integrand in the formula for the arc length s , we have that
1+ ((dy)/(dx))^2 = 1 + (4e^(2x))/((e^x+1)^2(e^x-1)^2) = ((e^x+1)^2(e^x-1)^2 + 4e^(2x))/((e^x+1)^2(e^x-1)^2)
and that
sqrt(1+((dy)/(dx))^2) = (sqrt((e^x+1)^2(e^x-1)^2 + 4e^(2x)))/((e^x+1)(e^x-1))
= sqrt((e^(2x)+2e^x +1)(e^(2x)-2e^(x)+1)+4e^(2x))/((e^x+1)(e^x-1))
= sqrt((e^(4x)-2e^(3x)+e^(2x)+2e^(3x) - 4e^(2x) + 2e^x + e^(2x)-2e^x + 1)+4e^(2x))/((e^x+1)(e^x-1))
Adding up that very long set of exponential terms, finding that most cancel, we finally have that
sqrt(1+((dy)/(dx))^2) = sqrt((e^(4x)-2e^(2x)+1)+4e^(2x))/((e^x+1)(e^x-1)) = sqrt(e^(4x)+2e^(2x)+1)/((e^x+1)(e^x-1))
= sqrt((e^(2x)+1)^2)/((e^x+1)(e^x-1)) = (e^(2x)+1)/((e^x+1)(e^x-1))
= (e^(2x)+1)/(e^(2x)-1)
Integrating the integrand over the required interval, we have that
s = int_(ln2)^(ln3) ((e^(2x)+1)/(e^(2x)-1)) \quad dx
= int_(ln2)^(ln3) (2e^(2x)- e^(2x) + 1)/(e^(2x)-1) \quad dx = int_(ln2)^(ln3) (2e^(2x))/(e^(2x)-1) -1 \quad dx
= ln(e^(2x)-1)|_ln2^(ln3) - x |_(ln2)^(ln3)
= ln (e^(2ln3)-1) - ln(e^(2ln2)-1) - ln3 + ln2
= ln(9-1) - ln(4-1) - ln3 + ln2 = ln((8/3)(2/3))
= ln(16/9) is the final answer