Question Number 81133 by 1406 last updated on 09/Feb/20
$${find}\:{this}\: \\ $$$$\int\frac{\mathrm{1}}{{sin}^{\mathrm{2}} {x}\sqrt[{\mathrm{7}}]{{cot}^{\mathrm{4}} {x}}}{dx} \\ $$
Commented by jagoll last updated on 10/Feb/20
$$\int\:\frac{{csc}^{\mathrm{2}} {x}\:{dx}}{\:\sqrt[{\mathrm{7}\:}]{\mathrm{cot}\:^{\mathrm{4}} {x}}}\:=\:−\int\frac{\:\:{d}\left(\mathrm{cot}\:{x}\right)}{\mathrm{cot}\:^{\frac{\mathrm{4}}{\mathrm{7}}} {x}}\:= \\ $$$$−\:\int\:{u}^{−\frac{\mathrm{4}}{\mathrm{7}}} \:{du}\:=\:−\frac{\mathrm{7}}{\mathrm{3}}\:\sqrt[{\mathrm{7}\:}]{\mathrm{cot}\:^{\mathrm{3}} {x}}\:+{c}\: \\ $$