Question Number 206248 by MetaLahor1999 last updated on 10/Apr/24
$$\int\frac{\mathrm{1}}{\:\sqrt{\left(\mathrm{1}−{t}\right)\left(\mathrm{2}−{t}\right)}}{dt}=...? \\ $$
Answered by TonyCWX08 last updated on 10/Apr/24
$$\int\frac{\mathrm{1}}{\:\sqrt{{t}^{\mathrm{2}} −\mathrm{3}{t}+\mathrm{2}}}{dt} \\ $$$$=\int\frac{\mathrm{1}}{\:\sqrt{\left({t}−\frac{\mathrm{3}}{\mathrm{2}}\right)^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{4}}}}{dt} \\ $$$${let} \\ $$$${u}\:=\:{t}−\frac{\mathrm{3}}{\mathrm{2}}\: \\ $$$${du}\:=\:{dx} \\ $$$$\int\:\frac{{du}}{\:\sqrt{{u}^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{4}}}} \\ $$$$={ln}\left(\mid{u}+\sqrt{{u}^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{4}}}\mid\right)+{c} \\ $$$$={ln}\left(\mid{t}−\frac{\mathrm{3}}{\mathrm{2}}+\sqrt{\left({t}−\frac{\mathrm{3}}{\mathrm{2}}\right)^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{4}}}\mid\right)+{c} \\ $$$$={ln}\left(\mid{t}−\frac{\mathrm{3}}{\mathrm{2}}+\sqrt{{t}^{\mathrm{2}} −\mathrm{3}{t}+\mathrm{2}}\mid\right)+{c} \\ $$$$ \\ $$
Commented by Frix last updated on 10/Apr/24
$$\mathrm{Yes}.\:\mathrm{But}\:\mathrm{for}\:\mathrm{the}\:\mathrm{less}\:\mathrm{advanced}\:\mathrm{it}\:\mathrm{would}\:\mathrm{be} \\ $$$$\mathrm{nice}\:\mathrm{to}\:\mathrm{show}\:\mathrm{the}\:\mathrm{path}\:\mathrm{for} \\ $$$$\int\frac{{du}}{\:\sqrt{{u}^{\mathrm{2}} \pm\mathrm{c}}}=\mathrm{ln}\:\mid{u}+\sqrt{{u}^{\mathrm{2}} \pm{c}}\mid \\ $$
Commented by TonyCWX08 last updated on 11/Apr/24
$${If}\:{you}\:{want},\:{sure}. \\ $$$${But}\:{it}\:{involves}\:{some}\:{trigonometry}\:{substitution}. \\ $$