Question Number 98831 by bramlex last updated on 16/Jun/20
Commented by john santu last updated on 16/Jun/20
$$\int\:\frac{\mathrm{2}\:\mathrm{dx}}{\mathrm{x}^{\mathrm{8}} \left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{7}} }\right)}\:=\:\int\:\frac{\mathrm{2d}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{7}} }\right)}{\mathrm{7}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{7}} }\right)} \\ $$$$=\:\frac{\mathrm{2}}{\mathrm{7}}\:\mathrm{ln}\:\left(\frac{\mathrm{x}^{\mathrm{7}} −\mathrm{1}}{\mathrm{x}^{\mathrm{7}} }\right)\:+\:\mathrm{c}\: \\ $$
Answered by Dwaipayan Shikari last updated on 17/Jun/20
$$\int\frac{\mathrm{2}}{{x}^{\mathrm{8}} −{x}}{dx}=\mathrm{2}\int\frac{\frac{\mathrm{1}}{{x}^{\mathrm{8}} }}{\mathrm{1}−\frac{\mathrm{1}}{{x}^{\mathrm{7}} }}{dx}=\frac{\mathrm{2}}{\mathrm{7}}{ln}\left(\mathrm{1}−\frac{\mathrm{1}}{{x}^{\mathrm{7}} }\right)+{constant} \\ $$