Question Number 49815 by Aditya789 last updated on 11/Dec/18
$$\mathrm{sin}^{\mathrm{6}} \mathrm{x}−\mathrm{cos}^{\mathrm{6}} \mathrm{x}/\mathrm{sin}^{\mathrm{2}} \mathrm{xcos}^{\mathrm{2}} \mathrm{x} \\ $$
Commented by Abdo msup. last updated on 12/Dec/18
$${if}\:{you}\:{mean}\:{simification} \\ $$$${A}\:=\frac{\left({sin}^{\mathrm{2}} {x}\right)^{\mathrm{3}} \:−\left({cos}^{\mathrm{2}} {x}\right)^{\mathrm{3}} }{{sin}^{\mathrm{2}} {x}\:.{cos}^{\mathrm{2}} {x}}\:=\frac{\left({sin}^{\mathrm{2}} {x}−{cos}^{\mathrm{2}} {x}\right)\left({sin}^{\mathrm{4}} {x}+{sin}^{\mathrm{2}} {xcos}^{\mathrm{2}} {x}\:+{cos}^{\mathrm{4}} {x}\right)}{{sin}^{\mathrm{2}} {x}\:.{cos}^{\mathrm{2}} {x}} \\ $$$$=\frac{−\mathrm{2}{cos}\left(\mathrm{2}{x}\right)\left(\:\left({sin}^{\mathrm{2}} {x}\:+{cos}^{\mathrm{2}} {x}\right)^{\mathrm{2}} −{sin}^{\mathrm{2}} {x}\:{cos}^{\mathrm{2}} {x}\right)}{{sin}^{\mathrm{2}} {x}\:{cos}^{\mathrm{2}} {x}} \\ $$$$=−\mathrm{2}{cos}\left(\mathrm{2}{x}\right)\left(\:\frac{\mathrm{4}}{{sin}^{\mathrm{2}} \left(\mathrm{2}{x}\right)}\:−\mathrm{1}\right) \\ $$$$=\frac{\mathrm{8}{cos}\left(\mathrm{2}{x}\right)}{{sin}^{\mathrm{2}} \left(\mathrm{2}{x}\right)}\:+\mathrm{2}{cos}\left(\mathrm{2}{x}\right)\:. \\ $$
Commented by Abdo msup. last updated on 12/Dec/18
$${A}\:=−\frac{\mathrm{8}{cos}\left(\mathrm{2}{x}\right)}{{sin}^{\mathrm{2}} \left(\mathrm{2}{x}\right)}\:+\mathrm{2}{cos}\left(\mathrm{2}{x}\right)\:. \\ $$
Commented by Abdo msup. last updated on 15/Dec/18
$$\int{A}\left({x}\right){dx}\:=−\mathrm{4}\:\int\:\:\:\frac{\mathrm{2}{cos}\left(\mathrm{2}{x}\right)}{{sin}^{\mathrm{2}} \left(\mathrm{2}{x}\right)}{dx}\:+\mathrm{2}\:\int\:{cos}\left(\mathrm{2}{x}\right){dx} \\ $$$$=\frac{\mathrm{4}}{{sin}\left(\mathrm{2}{x}\right)}\:+{sin}\left(\mathrm{2}{x}\right)+{c} \\ $$