Question Number 117666 by danielasebhofoh last updated on 13/Oct/20
Commented by bemath last updated on 13/Oct/20
$$\mathrm{set}\:\sqrt{\mathrm{ax}+\mathrm{b}}\:=\mathrm{t}\:\Rightarrow\:\mathrm{ax}+\mathrm{b}\:=\:\mathrm{t}^{\mathrm{2}} \\ $$$$\mathrm{x}\:=\:\frac{\mathrm{t}^{\mathrm{2}} −\mathrm{b}}{\mathrm{a}}\:\Rightarrow\mathrm{dx}\:=\:\frac{\mathrm{2t}\:\mathrm{dt}}{\mathrm{a}} \\ $$$$\int\:\frac{\mathrm{2t}\:\mathrm{dt}}{\mathrm{a}\left(\frac{\mathrm{t}^{\mathrm{2}} −\mathrm{b}}{\mathrm{a}}\:+\:\mathrm{c}\right).\mathrm{t}}\:=\:\int\:\frac{\mathrm{2}\:\mathrm{dt}}{\mathrm{t}^{\mathrm{2}} −\left(\mathrm{b}−\mathrm{ac}\right)} \\ $$$$\int\:\frac{\mathrm{2}\:\mathrm{dt}}{\left(\mathrm{t}−\sqrt{\mathrm{b}−\mathrm{ac}}\right)\left(\mathrm{t}+\sqrt{\mathrm{b}−\mathrm{ac}}\right)}\: \\ $$$$\mathrm{now}\:\mathrm{it}\:\mathrm{easy}\:\mathrm{to}\:\mathrm{solve} \\ $$
Commented by MJS_new last updated on 13/Oct/20
$$\mathrm{typo}:\:\int\frac{\mathrm{2}{dt}}{{t}^{\mathrm{2}} −\left({b}−{ac}\right)} \\ $$
Commented by bemath last updated on 13/Oct/20
$$\mathrm{haha}..\mathrm{yes}\:\mathrm{sir}\:\mathrm{thank}\:\mathrm{you} \\ $$