Question Number 167006 by cortano1 last updated on 04/Mar/22
$$\:\:\:\:\:\:\:\int\:\frac{\mathrm{3x}^{\mathrm{3}} }{\left(\mathrm{x}−\mathrm{1}\right)^{\mathrm{3}} }\:\mathrm{dx}=? \\ $$
Answered by MJS_new last updated on 04/Mar/22
$$\int\frac{\mathrm{3}{x}^{\mathrm{3}} }{\left({x}−\mathrm{1}\right)^{\mathrm{3}} }{dx}=\int\frac{\mathrm{3}\left({t}+\mathrm{1}\right)^{\mathrm{3}} }{{t}^{\mathrm{3}} }{dt}= \\ $$$$=\int\left(\mathrm{3}+\frac{\mathrm{9}}{{t}}+\frac{\mathrm{9}}{{t}^{\mathrm{2}} }+\frac{\mathrm{3}}{{t}^{\mathrm{3}} }\right){dt}= \\ $$$$=\mathrm{3}{t}+\mathrm{9ln}\:{t}\:−\frac{\mathrm{9}}{{t}}−\frac{\mathrm{3}}{\mathrm{2}{t}^{\mathrm{2}} }= \\ $$$$=\frac{\mathrm{2}\left(\mathrm{2}{t}^{\mathrm{3}} −\mathrm{6}{t}−\mathrm{1}\right)}{\mathrm{2}{t}^{\mathrm{2}} }+\mathrm{9ln}\:{t}\:= \\ $$$$=\frac{\mathrm{3}\left(\mathrm{2}{x}^{\mathrm{3}} −\mathrm{6}{x}^{\mathrm{2}} +\mathrm{3}\right)}{\mathrm{2}\left({x}−\mathrm{1}\right)^{\mathrm{2}} }+\mathrm{9ln}\:\mid{x}−\mathrm{1}\mid\:+{C} \\ $$
Commented by peter frank last updated on 04/Mar/22
$$\mathrm{thank}\:\mathrm{you} \\ $$