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Question Number 105951 by Study last updated on 01/Aug/20

Answered by Dwaipayan Shikari last updated on 01/Aug/20

(√(x/( (√(x/((x/)...))))))=p  (√(x/p))=p  p=(x)^(1/3)   ∫pdx=∫x^(1/3) dx=(3/4)x^(4/3) +C

$$\sqrt{\frac{{x}}{\:\sqrt{\frac{{x}}{\frac{{x}}{}...}}}}={p} \\ $$$$\sqrt{\frac{{x}}{{p}}}={p} \\ $$$${p}=\sqrt[{\mathrm{3}}]{{x}} \\ $$$$\int{pdx}=\int{x}^{\frac{\mathrm{1}}{\mathrm{3}}} {dx}=\frac{\mathrm{3}}{\mathrm{4}}{x}^{\frac{\mathrm{4}}{\mathrm{3}}} +{C} \\ $$

Commented by malwaan last updated on 02/Aug/20

(√x) = (√(x/(√(x/.))))   is this right ?

$$\sqrt{{x}}\:=\:\sqrt{\frac{{x}}{\sqrt{\frac{{x}}{.}}}}\: \\ $$$${is}\:{this}\:{right}\:? \\ $$

Commented by Her_Majesty last updated on 03/Aug/20


$${yes} \\ $$

Answered by Her_Majesty last updated on 01/Aug/20

y=(√(x/(√(x/(...)))))  ⇒  y^2 =(x/y)  ⇒  y=x^(1/3)   ∫x^(1/3) dx=(3/4)x^(4/3) +C

$${y}=\sqrt{\frac{{x}}{\sqrt{\frac{{x}}{...}}}} \\ $$$$\Rightarrow \\ $$$${y}^{\mathrm{2}} =\frac{{x}}{{y}} \\ $$$$\Rightarrow \\ $$$${y}={x}^{\frac{\mathrm{1}}{\mathrm{3}}} \\ $$$$\int{x}^{\frac{\mathrm{1}}{\mathrm{3}}} {dx}=\frac{\mathrm{3}}{\mathrm{4}}{x}^{\frac{\mathrm{4}}{\mathrm{3}}} +{C} \\ $$

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