Question Number 107589 by mathdave last updated on 11/Aug/20
Answered by Ar Brandon last updated on 11/Aug/20
$$\mathcal{I}=\int\left(\frac{\mathrm{x}+\mathrm{6}}{\mathrm{x}+\mathrm{8}}\right)^{\mathrm{6}} \mathrm{dx}=\int\left(\mathrm{1}−\frac{\mathrm{2}}{\mathrm{x}+\mathrm{8}}\right)^{\mathrm{6}} \mathrm{dx} \\ $$$$\:\:\:=\int\left\{\mathrm{1}−\frac{\mathrm{12}}{\mathrm{x}+\mathrm{8}}+\frac{\mathrm{60}}{\left(\mathrm{x}+\mathrm{8}\right)^{\mathrm{2}} }−\frac{\mathrm{160}}{\left(\mathrm{x}+\mathrm{8}\right)^{\mathrm{3}} }+\frac{\mathrm{240}}{\left(\mathrm{x}+\mathrm{8}\right)^{\mathrm{4}} }−\frac{\mathrm{192}}{\left(\mathrm{x}+\mathrm{8}\right)^{\mathrm{5}} }+\frac{\mathrm{64}}{\left(\mathrm{x}+\mathrm{8}\right)^{\mathrm{6}} }\right\}\mathrm{dx} \\ $$$$\:\:\:=\mathrm{x}−\mathrm{12ln}\mid\mathrm{x}+\mathrm{8}\mid−\frac{\mathrm{60}}{\mathrm{x}+\mathrm{8}}+\frac{\mathrm{80}}{\left(\mathrm{x}+\mathrm{8}\right)^{\mathrm{2}} }−\frac{\mathrm{80}}{\left(\mathrm{x}+\mathrm{8}\right)^{\mathrm{3}} }+\frac{\mathrm{48}}{\left(\mathrm{x}+\mathrm{8}\right)^{\mathrm{4}} }−\frac{\mathrm{12},\mathrm{8}}{\left(\mathrm{x}+\mathrm{8}\right)^{\mathrm{5}} }+\mathcal{C} \\ $$
Answered by Her_Majesty last updated on 11/Aug/20
$${t}={x}+\mathrm{8} \\ $$$$\int\frac{\left({t}−\mathrm{2}\right)^{\mathrm{6}} }{{t}^{\mathrm{6}} }{dt}= \\ $$$$=−\mathrm{12}{ln}\mid{t}\mid+{t}−\frac{\mathrm{60}}{{t}}+\frac{\mathrm{80}}{{t}^{\mathrm{2}} }−\frac{\mathrm{80}}{{t}^{\mathrm{3}} }+\frac{\mathrm{48}}{{t}^{\mathrm{4}} }−\frac{\mathrm{64}}{\mathrm{5}{t}^{\mathrm{5}} }= \\ $$$$=−\mathrm{12}{ln}\mid{x}+\mathrm{8}\mid−\frac{\mathrm{5}{x}^{\mathrm{6}} +\mathrm{240}{x}^{\mathrm{5}} +\mathrm{4500}{x}^{\mathrm{4}} +\mathrm{42000}{x}^{\mathrm{3}} +\mathrm{201200}{x}^{\mathrm{2}} +\mathrm{439280}{x}+\mathrm{262976}}{\mathrm{5}\left({x}+\mathrm{8}\right)^{\mathrm{5}} }+{C} \\ $$