Answer
$\ln(\frac{2+\sqrt 3}{\sqrt 2+1})$
Work Step by Step
$\frac{dy}{dx}$ = $-cot(x)$
$L$ = $\int_{{\,\frac{\pi}{6}}}^{{\,\frac{\pi}{4}}}$$\sqrt{1+(-cot(x))^{2}}$ $dx$
$L$ = $\int_{{\,\frac{\pi}{6}}}^{{\,\frac{\pi}{4}}}$$\sqrt{1+cot^{2}(x)}$ $dx$
$L$ = $\int_{{\,\frac{\pi}{6}}}^{{\,\frac{\pi}{4}}}$$\sqrt{csx^{2}(x)}$ $dx$
$L$ = $\int_{{\,\frac{\pi}{6}}}^{{\,\frac{\pi}{4}}}$${csx(x)}$ $dx$
$L$ = $-\ln|{csx(x)+cot(x)}|$ $|_{{\,\frac{\pi}{6}}}^{{\,\frac{\pi}{4}}}$
$L$ = $-[(\ln|csx(\frac{\pi}{4})+cot(\frac{\pi}{4})|)-(\ln|{csx(\frac{\pi}{6})+cot(\frac{\pi}{6})}|)]$
$L$ = $[-(\ln|\sqrt 2+1|)+(\ln|2+\sqrt 3|)]$
$L$ = $\ln(\frac{2+\sqrt 3}{\sqrt 2+1})$
