Question Number 209662 by depressiveshrek last updated on 18/Jul/24
$$\int\frac{\mathrm{1}}{\mathrm{sin}^{\mathrm{2}} {x}\left(\mathrm{1}+\mathrm{cos}^{\mathrm{2}} {x}\right)}{dx} \\ $$
Answered by ARUNG_Brandon_MBU last updated on 18/Jul/24
$${I}=\int\frac{{dx}}{\mathrm{sin}^{\mathrm{2}} {x}\left(\mathrm{1}+\mathrm{cos}^{\mathrm{2}} {x}\right)} \\ $$$$\:\:=\int\frac{\mathrm{tan}^{\mathrm{2}} {x}+\mathrm{1}}{\mathrm{tan}^{\mathrm{2}} {x}\left(\mathrm{tan}^{\mathrm{2}} {x}+\mathrm{2}\right)}\left(\mathrm{sec}^{\mathrm{2}} {xdx}\right) \\ $$$$\:\:=\frac{\mathrm{1}}{\mathrm{2}}\int\frac{\left(\mathrm{tan}^{\mathrm{2}} {x}+\mathrm{2}\right)+\mathrm{tan}^{\mathrm{2}} {x}}{\mathrm{tan}^{\mathrm{2}} {x}\left(\mathrm{tan}^{\mathrm{2}} {x}+\mathrm{2}\right)}{d}\left(\mathrm{tan}{x}\right) \\ $$$$\:\:=−\frac{\mathrm{1}}{\mathrm{2tan}{x}}+\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{2}}}\mathrm{arctan}\left(\frac{\mathrm{tan}{x}}{\:\sqrt{\mathrm{2}}}\right)+{C} \\ $$