Question Number 191188 by cortano12 last updated on 20/Apr/23
$$\:\:\:\:\:\:\:\:\:\:\:\int\:\sqrt[{\mathrm{4}}]{\frac{\mathrm{2}−\mathrm{x}}{\mathrm{1}−\mathrm{x}}}\:\mathrm{dx}\:=?\: \\ $$
Answered by mehdee42 last updated on 20/Apr/23
$$\sqrt[{\mathrm{4}}]{\frac{\mathrm{2}−{x}}{\mathrm{1}−{x}}}={u}\Rightarrow{x}=\frac{{u}^{\mathrm{4}} −\mathrm{2}}{{u}^{\mathrm{4}} −\mathrm{1}}\Rightarrow{dx}=\frac{\mathrm{4}{u}^{\mathrm{3}} }{\left({u}^{\mathrm{4}} −\mathrm{1}\right)^{\mathrm{2}} \:}\:{du} \\ $$$$\Rightarrow\int\sqrt[{\mathrm{4}}]{\frac{{x}−\mathrm{2}}{{x}−\mathrm{1}}}{dx}=\int\frac{\mathrm{4}{u}^{\mathrm{4}} }{\left({u}^{\mathrm{4}} −\mathrm{1}\right)^{\mathrm{2}} }{du} \\ $$$${this}\:{integral}\:{can}\:{be}\:{solved}\:{by}\:{dividing}\:{fractions}. \\ $$