[tex]\displaystyle I = \int_{\frac{\pi}{4}}^\frac{\pi}{2}(\mathrm{ctg}^2 x+ \mathrm{ctg}^4 x+ \mathrm{ctg}^6 x+\mathrm{ctg}^8 x)\, dx = \\ \\ = \int_{\frac{\pi}{4}}^\frac{\pi}{2}\Big[\mathrm{ctg^2}x(1+ \mathrm{ctg}^2 x)+ \mathrm{ctg}^6 x(1+\mathrm{ctg}^2 x)\Big]\, dx= \\ \\= \int_{\frac{\pi}{4}}^\frac{\pi}{2}(1+\mathrm{ctg^2}x)(\mathrm{ctg^2}x+\mathrm{ctg^6}x)\, dx \\ \\ \\\mathrm{ctg}\, x = t \Rightarrow \mathrm{tg}^{-1}x = t \Rightarrow -\mathrm{tg}^{-2} x \cdot (\mathrm{tg}^2 x+1)\, dx = dt \Rightarrow[/tex]
[tex]\Rightarrow -(1+\mathrm{ctg}^2 x)\, dx = dt \Rightarrow (1+\mathrm{ctg}^2 x)\, dx = -dt\\ \\ x = \dfrac{\pi}{4} \Rightarrow t = 1,\quad x = \dfrac{\pi}{2}\Rightarrow t = 0 \\ \\ I = \displaystyle \int_{1}^0 (t^2+ t^6)\cdot (-dt) = \int_{0}^1(t^2+t^6)\, dt = \\ \\ =\Big(\dfrac{t^3}{3}+\dfrac{t^7}{7}\Big)\Bigg|_{0}^1 = \dfrac{10}{21}[/tex]
