Question Number 108169 by bemath last updated on 15/Aug/20
$$\:\:\:\:\:\:\frac{\checkmark\mathcal{B}{e}\mathcal{M}{ath}\checkmark}{\succ\prec} \\ $$$$\int\:\frac{{x}^{\mathrm{5}} −{x}}{{x}^{\mathrm{8}} −\mathrm{1}}\:{dx}\:? \\ $$
Answered by Dwaipayan Shikari last updated on 15/Aug/20
$$\int\frac{{x}\left({x}^{\mathrm{4}} −\mathrm{1}\right)}{\left({x}^{\mathrm{8}} −\mathrm{1}\right)}{dx} \\ $$$$\int\frac{{x}}{{x}^{\mathrm{4}} +\mathrm{1}}{dx}\:\:\:\:\:\:{x}^{\mathrm{2}} ={u}\:\:\:,\mathrm{2}{x}=\frac{{du}}{{dx}} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\int\frac{{du}}{{u}^{\mathrm{2}} +\mathrm{1}}=\frac{\mathrm{1}}{\mathrm{2}}{tan}^{−\mathrm{1}} {u}+{C}=\frac{\mathrm{1}}{\mathrm{2}}{tan}^{−\mathrm{1}} \left({x}^{\mathrm{2}} \right)+{C} \\ $$