Find the arc length of the given function/curve on the given interval.

y=ln(x-sqrt(x^(2)-1)); x ϵ [1, sqrt(2)]

y = ln(x-√(x^2-1))

y' = 1/x-√(x^2-1) * (1-x/√(x^2-1))
= -1/√(x^2-1)

so, the arc length is

∫[1,√2] √(1+y'^2) dx
= ∫[1,√2] √(1+1/(x^2-1)) dx
= ∫[1,√2] x/√(x^2-1) dx
. . .

To find the arc length of the given function, we can use the formula for arc length of a curve:

L = ∫[a,b] sqrt(1 + (dy/dx)^2) dx

In this case, the function is y = ln(x - sqrt(x^2 - 1)) and the interval is x ∈ [1, sqrt(2)].

Step 1: Find the derivative of y with respect to x
We have:

dy/dx = d/dx(ln(x - sqrt(x^2 - 1)))

To find the derivative, we can use the chain rule. Let u = x - sqrt(x^2 - 1), then du/dx = 1 - (1/2)(2x)(1/sqrt(x^2 - 1)) = 1 - x/sqrt(x^2 - 1). Applying the chain rule:

dy/dx = 1/u * du/dx = (1 - x/sqrt(x^2 - 1))/u

Step 2: Simplify the expression
Recall that u = x - sqrt(x^2 - 1), so we can substitute this back into the expression for dy/dx:

dy/dx = (1 - x/sqrt(x^2 - 1))/(x - sqrt(x^2 - 1))

Step 3: Find the integrand
The integrand in the arc length formula is sqrt(1 + (dy/dx)^2).

Plugging in the expression for dy/dx, we have:

Integrand = sqrt(1 + ((1 - x/sqrt(x^2 - 1))/(x - sqrt(x^2 - 1)))^2)

Step 4: Find the integral
Now we can find the integral of the integrand over the interval [1, sqrt(2)]. This will give us the arc length L.

L = ∫[1, sqrt(2)] sqrt(1 + ((1 - x/sqrt(x^2 - 1))/(x - sqrt(x^2 - 1)))^2) dx

Unfortunately, this integral does not have a simple closed-form solution, so it cannot be solved analytically. Instead, you would need numerical methods such as approximation techniques or software to evaluate the integral and find the arc length.

To find the arc length of a function or curve, we can use the arc length formula:

L = ∫[a, b] sqrt(1 + (dy/dx)^2) dx

In this case, we have the function y = ln(x - sqrt(x^2 - 1)), where x is in the interval [1, sqrt(2)].

To find dy/dx, we need to take the derivative of y with respect to x:

dy/dx = d/dx[ln(x - sqrt(x^2 - 1))]

To simplify this, let's rewrite the function in a slightly different form first:

y = ln(x - sqrt(x^2 - 1))
y = ln(x - (x^2 - 1)^0.5)

Now, we can take the derivative:

dy/dx = d/dx[ln(x - (x^2 - 1)^0.5)]
= d/dx[ln(x - (x^2 - 1)^0.5)]

Using the chain rule, we can differentiate the natural logarithm:

dy/dx = 1/(x - (x^2 - 1)^0.5) * d/dx[x - (x^2 - 1)^0.5]
= 1/(x - (x^2 - 1)^0.5) * (1 - 0.5(2x)(x^2 - 1)^-0.5)
= 1/(x - (x^2 - 1)^0.5) * (1 - x(x^2 - 1)^-0.5)

Now that we have dy/dx, we can substitute it back into the arc length formula and integrate. However, before doing so, we should simplify the expression inside the square root to make it easier to integrate:

(x^2 - 1)^0.5 = (x^2 - 1)^(1/2) = sqrt(x^2 - 1)

Now, let's substitute dy/dx and simplify the arc length formula:

L = ∫[1, sqrt(2)] sqrt(1 + (dy/dx)^2) dx
= ∫[1, sqrt(2)] sqrt(1 + (1/(x - sqrt(x^2 - 1)))^2) dx
= ∫[1, sqrt(2)] sqrt(1 + (1/(x - sqrt(x^2 - 1)))^2) dx

Unfortunately, the integral does not have a neat closed-form solution and cannot be expressed in terms of elementary functions. Therefore, we would need to use numerical methods, such as numerical integration or software, to find an approximate value for the arc length.