Question Number 76424 by john santu last updated on 27/Dec/19
$${how}\:{to}\:{calculate}\:\int\sqrt{\frac{{x}}{\:\sqrt{\frac{{x}}{\:\sqrt{\frac{{x}}{\:\sqrt{…}}}}}}}\:\:{dx}\:? \\ $$
Commented by john santu last updated on 27/Dec/19
$${this}\:{the}\:{crazy}\:{integration} \\ $$
Answered by $@ty@m123 last updated on 27/Dec/19
$${y}=\sqrt{\frac{{x}}{{y}}} \\ $$$${y}^{\mathrm{2}} =\frac{{x}}{{y}} \\ $$$${y}^{\mathrm{3}} ={x} \\ $$$${y}={x}^{\frac{\mathrm{1}}{\mathrm{3}}} \\ $$$$\therefore\:{the}\:{given}\:{integral} \\ $$$$=\int{x}^{\frac{\mathrm{1}}{\mathrm{3}}} {dx} \\ $$$$=\frac{\mathrm{3}}{\mathrm{4}}{x}^{\frac{\mathrm{4}}{\mathrm{3}}} +\mathrm{C} \\ $$
Commented by john santu last updated on 27/Dec/19
$${thanks}\:{you}\:{sir} \\ $$