Answer
$\approx{1.767}$
Work Step by Step
The formula for the arc length is
$s=\int_{a}^b \sqrt{1 + (y')^2} dy$
Differentiate $y = \ln{\sin{x}}$ with respect to x.
$y' = \frac{\cos{x}}{\sin{x}}$
$=\cot{x}$
Substitute the value of y' in $1 + (y')^2$
$1+(y')^2=1+{\cot{^2}{x}}$
$=\csc{^2}{x}$
Substitute the value of $1 + (y')^2$ in the formula for the arc length $s=\int_{a}^{b} \sqrt{1 + (y')^2} dy$
Solve for the distance s
$s=\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}}\sqrt{\csc{^2}{x}} dx$
$\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}}{\csc{x}} dx$
$=[-\ln\lvert \csc{x}+\cot{x} \rvert]_{\frac{\pi}{4}}^{\frac{3\pi}{4}}\ $
$\approx{1.767}$
