Question Number 20166 by tammi last updated on 23/Aug/17
$$\int\mathrm{cosec}\:^{\mathrm{2}} {xdx} \\ $$
Answered by sma3l2996 last updated on 23/Aug/17
$${t}={cosecx}\Rightarrow{dt}=−\frac{{cosec}^{\mathrm{2}} {x}}{{secx}}{dx} \\ $$$${cosec}^{\mathrm{2}} {xdx}=−\frac{{tdt}}{\:\sqrt{{t}^{\mathrm{2}} −\mathrm{1}}} \\ $$$$\int{cosec}^{\mathrm{2}} {xdx}=−\int\frac{{tdt}}{\:\sqrt{{t}^{\mathrm{2}} −\mathrm{1}}}=−\int\frac{{d}\left({t}^{\mathrm{2}} −\mathrm{1}\right)}{\mathrm{2}\sqrt{{t}^{\mathrm{2}} −\mathrm{1}}} \\ $$$$=−\sqrt{{t}^{\mathrm{2}} −\mathrm{1}}+{C}=−\sqrt{{cosec}^{\mathrm{2}} {x}−\mathrm{1}}+{C} \\ $$$$\int{cosec}^{\mathrm{2}} {xdx}=−{cotx}+{C} \\ $$