Hello everyone. Much to my chagrin, I am having an algebraic problem. Here is the question:

Question:

Find the length of the curve analytically by anti-differentiation. You will need to simplify the integrand algebraically before finding an anti-derivative.

$\displaystyle y = \frac{x^3}{3} + x^2 + x + \frac{1}{4x + 4}, 0 \leq x \leq 2 $

Solution:

$\displaystyle L = \int_{a}^{b} \sqrt{1 + (\frac{dy}{dx})^2} dx $

$\displaystyle \frac{dy}{dx} = x^2 + 2x + 1 - \frac{1}{4(x + 1)^2} = (x+1)^2 - \frac{1}{4(x+1)^2} $

$\displaystyle L = \int_{0}^{2} \sqrt{1 + ( (x+1)^2 - \frac{1}{4(x+1)^2} )^2} $

I understand everything up to this point. In the answer, this is what it does after the prior step:

$\displaystyle = \int_{0}^{2} \sqrt{( (x+1)^2 + \frac{1}{4(x+1)^2} )^2} $

What happened to the 1? What happened to the negative sign? Any help is appreciated. Thanks in advance.