Question Number 108325 by john santu last updated on 16/Aug/20
$$\:\:\:\frac{\multimap{JS}\multimap}{−−−−} \\ $$$$\int\:\frac{{dx}}{{x}^{\mathrm{8}} \left({x}^{\mathrm{2}} +\mathrm{1}\right)}\:=\:? \\ $$
Commented by john santu last updated on 16/Aug/20
$$\:\:\:\frac{\multimap{JS}\multimap}{\_\_\_\_\_\_} \\ $$$$\int\:\frac{{dx}}{{x}^{\mathrm{8}} \left({x}^{\mathrm{2}} +\mathrm{1}\right)}\:=\:\int\left(\frac{\mathrm{1}}{{x}^{\mathrm{2}} +\mathrm{1}}−\frac{\mathrm{1}}{{x}^{\mathrm{2}} }+\frac{\mathrm{1}}{{x}^{\mathrm{4}} }−\frac{\mathrm{1}}{{x}^{\mathrm{6}} }+\frac{\mathrm{1}}{{x}^{\mathrm{8}} }\right){dx} \\ $$$$=\:\mathrm{tan}^{−\mathrm{1}} \left({x}\right)+\frac{\mathrm{1}}{{x}}−\frac{\mathrm{1}}{\mathrm{3}{x}^{\mathrm{3}} }+\frac{\mathrm{1}}{\mathrm{5}{x}^{\mathrm{5}} }−\frac{\mathrm{1}}{\mathrm{7}{x}^{\mathrm{7}} }\:+\:{c} \\ $$