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

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