find the exact length of the curve

y = ln(1-x^2), 0 <= x <= (1/2)

So by doing the work I eventually get down to

integral from 0 to 1/2 (1 + x^2)/(1-x^2) dx

but I keep getting

-1 + 1/(x-1) - 1/(x+1) after partial fractions which gives the answer

ln(1/3) - 1/2 when I evaluate the integral

the answer is ln(3) - 1/2 and it seems I am screwing up during the partial fraction. What exactly is wrong here?

the length of curve

= integral [ √(1 + (dy/dx)^2) from 0 to 1/2

dy/dx = -2x/(1-x^2) = 2x/(x^2-1)
which I separated into
1/(x+1) + 1/(x-1) using partial fractions.

we now have to square this, add it to one and take the square root.
Perhaps you tried to split it into partial fractions after you squared the derivative, I did it before.

integral [ √( 1 + ( 1/(x+1) + 1/(x-1) )^2 ) ]

does that help so far?

Wolfram says this:
http://integrals.wolfram.com/index.jsp?expr=√%281+%2B+%282x%2F%28x%5E2-1%29%29%5E2%29&random=false

To find the exact length of the curve defined by the equation y = ln(1-x^2) over the interval 0 ≤ x ≤ 1/2, you have correctly determined that you need to evaluate the integral

∫₀^(1/2) (1 + x^2)/(1 - x^2) dx.

The integrand can be rewritten using the method of partial fractions. Here's how to do it:

1. Factor the denominator: 1 - x^2 = (1 - x)(1 + x).

2. Write the integrand as a sum of partial fractions:

(1 + x^2)/(1 - x^2) = A/(1 - x) + B/(1 + x).

3. Find the common denominator on the right side:

A(1 + x) + B(1 - x) = 1 + x^2.

4. Next, equate the corresponding coefficients of the numerators on both sides:

For the constant term (1): A + B = 1.
For the coefficient of x term (0): A - B = 0.

5. Solve the system of equations to find the values of A and B. In this case, you'll find A = 1/2 and B = 1/2.

6. Substitute the values of A and B back into the partial fraction decomposition:

(1 + x^2)/(1 - x^2) = 1/2/(1 - x) + 1/2/(1 + x).

Now, you can rewrite the integral as:

∫₀^(1/2) 1/2/(1 - x) + 1/2/(1 + x) dx.

By evaluating this integral correctly, you should arrive at the desired answer of ln(3) - 1/2.

It's possible that there may have been a mistake made during the partial fraction decomposition or subsequent integration steps. Double-check your calculations to ensure accuracy.