Question Number 86080 by ar247 last updated on 27/Mar/20
$$\int\frac{{dx}}{\:\sqrt{\mathrm{5}−\mathrm{4}{x}−\mathrm{2}{x}^{\mathrm{2}} }} \\ $$
Commented by ar247 last updated on 27/Mar/20
$${anyone}\:{can}\:{help}\:{me}\:{please}? \\ $$
Answered by Rio Michael last updated on 27/Mar/20
$$\mathrm{5}−\mathrm{4}{x}−\mathrm{2}{x}^{\mathrm{2}} \:=\:\left(\frac{\mathrm{5}}{\mathrm{2}}\:−\mathrm{2}{x}−{x}^{\mathrm{2}} \right)\mathrm{2} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\left(\frac{\mathrm{5}}{\mathrm{2}}−\mathrm{1}\:+\:\left(\mathrm{1}−{x}\right)^{\mathrm{2}} \right)\mathrm{2} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\left(\frac{\mathrm{3}}{\mathrm{2}}\:+\:\left(\mathrm{1}−{x}\right)^{\mathrm{2}} \right)\mathrm{2} \\ $$$${let}\:\mathrm{1}−{x}\:=\:{u}\:\Rightarrow\:−{du}=\:{dx} \\ $$$$\:\int\frac{{dx}}{\:\sqrt{\mathrm{5}−\mathrm{4}{x}−\mathrm{2}{x}^{\mathrm{2}} }}\:=\:−\frac{\mathrm{1}}{\mathrm{2}}\int\frac{{du}}{\:\sqrt{\frac{\mathrm{3}}{\mathrm{2}}\:+\:{u}^{\mathrm{2}} }}\:=\:−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{arsinh}\left(\frac{\sqrt{\mathrm{2}}{x}}{\:\sqrt{\mathrm{3}}}\right)\:+\:{k} \\ $$$$\: \\ $$
Commented by ar247 last updated on 27/Mar/20
$${anything}\:{elese}? \\ $$