Question

$\int\limits^\frac{\pi }{4} _\frac{\pi }{6} {(tgx +ctgx)} ^{-1} \, dx$

Answer 1

$\int_{\frac{\pi}{6}}^{\frac{\pi}{4}}\frac{dx}{tgx+ctgx}=\int_{\frac{\pi}{6}}^{\frac{\pi}{4}}\left (cosx\cdot sinx \right )dx=\left.\begin{matrix}\frac{sin^2x}{2}\end{matrix}\right|_{\frac{\pi}{6}}^{\frac{\pi}{4}}=\frac{1}{8}$