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1-e-dx-x-ln-x-1-3-




Question Number 128279 by liberty last updated on 06/Jan/21
  Θ = ∫_1 ^( e)  (dx/(x.((ln x))^(1/3) )) ?
$$\:\:\Theta\:=\:\int_{\mathrm{1}} ^{\:\mathrm{e}} \:\frac{{dx}}{{x}.\sqrt[{\mathrm{3}}]{\mathrm{ln}\:{x}}}\:?\: \\ $$
Answered by john_santu last updated on 06/Jan/21
 let ((ln x))^(1/(3 ))  = w ⇒ (dx/x) = 3w^2  dw   Θ = ∫_0 ^( 1)  ((3w^2 )/w) dw = ∫_0 ^( 1)  3w dw     Θ = (3/2)w^2  ∣_0 ^1  = (3/2) .
$$\:{let}\:\sqrt[{\mathrm{3}\:}]{\mathrm{ln}\:{x}}\:=\:{w}\:\Rightarrow\:\frac{{dx}}{{x}}\:=\:\mathrm{3}{w}^{\mathrm{2}} \:{dw} \\ $$$$\:\Theta\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\mathrm{3}{w}^{\mathrm{2}} }{{w}}\:{dw}\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\mathrm{3}{w}\:{dw}\:\: \\ $$$$\:\Theta\:=\:\frac{\mathrm{3}}{\mathrm{2}}{w}^{\mathrm{2}} \:\mid_{\mathrm{0}} ^{\mathrm{1}} \:=\:\frac{\mathrm{3}}{\mathrm{2}}\:. \\ $$

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