Question Number 163888 by zakirullah last updated on 11/Jan/22
$${prove}\int\frac{\mathrm{1}−{x}^{\mathrm{2}} }{\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}{dx}\:=\:\int\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }{dx} \\ $$
Commented by mr W last updated on 12/Jan/22
$$\frac{{a}^{\mathrm{2}} }{{a}}={a}\:{is}\:{obvious}. \\ $$
Answered by essojean last updated on 11/Jan/22
$$\int\frac{\mathrm{1}−{x}^{\mathrm{2}} }{\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}{dx}=\int\left(\mathrm{1}−{x}^{\mathrm{2}} \right)×\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\frac{−\mathrm{1}}{\mathrm{2}}} {dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\int\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}} {dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\int\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\frac{\mathrm{1}}{\mathrm{2}}} {dx} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\int\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }{dx}..... \\ $$$$ \\ $$
Commented by zakirullah last updated on 12/Jan/22
$${nice}\:{solution}\:{Dear}\:{Sir} \\ $$$${A}\:{boundle}\:{of}\:{thanks}. \\ $$