Find the length of \( y= \ln x - \frac{x^2}{8} \) , \( 1 \leq x \leq 2 \)

\( \frac{dy}{dx}=\frac{1}{x} - \frac{x}{4} \)
\( \left ( \frac{dy}{dx} \right ) ^2 = \frac{1}{x^2}-\frac{1}{2} + \frac{x^2}{16} \)
\( 1+ \left ( \frac{dy}{dx} \right ) ^2 = \frac{1}{x^2}+\frac{1}{2} + \frac{x^2}{16} = \left ( \frac{1}{x} + \frac{x}{4} \right )^2 \)

\( S = \int \limits _1 ^2 \sqrt{\left ( \frac{1}{x} + \frac{x}{4} \right )^2} dx = \int \limits _1 ^2 \left ( \frac{1}{x} + \frac{x}{4} \right ) dx = \left [ \ln x + \frac{x^2}{8} \right ] _1 ^2 \)
\( = \left ( \ln 2 + \frac{1}{2} \right ) - \left ( \ln 1 + \frac{1}{8} \right ) \)
\( = \left ( \ln 2 + \frac{3}{8} \right ) \)