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Question Number 147444 by aliibrahim1 last updated on 20/Jul/21

Answered by SEKRET last updated on 21/Jul/21

∫ x^(1/2) ∙x^(1/(2∙3)) ∙x^(1/(2∙3∙4)) ∙....dx= ∫ x^((1/2)+(1/(2∙3))+(1/(2∙3∙4))+) ... dx= ∫ x^((1/(2!))+(1/(3!))+(1/(4!))+..) dx n=1 →(n/(0!))+ (n/(1!))+(n/(2!))+(n/(3!))+...= e^n (1/(2!))+(1/(3!))+(1/(4!))+.. = e−2 ∫ x^(e−2) dx = (x^(e−1) /(e−1)) +C ABDULAZIZ ABDUVALIYEV

$$\int\:\boldsymbol{\mathrm{x}}^{\frac{\mathrm{1}}{\mathrm{2}}} \centerdot\boldsymbol{\mathrm{x}}^{\frac{\mathrm{1}}{\mathrm{2}\centerdot\mathrm{3}}} \centerdot\boldsymbol{\mathrm{x}}^{\frac{\mathrm{1}}{\mathrm{2}\centerdot\mathrm{3}\centerdot\mathrm{4}}} \centerdot....\boldsymbol{\mathrm{dx}}= \\ $$$$\int\:\boldsymbol{\mathrm{x}}^{\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}\centerdot\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{2}\centerdot\mathrm{3}\centerdot\mathrm{4}}+} ...\:\mathrm{d}\boldsymbol{\mathrm{x}}=\:\int\:\boldsymbol{\mathrm{x}}^{\frac{\mathrm{1}}{\mathrm{2}!}+\frac{\mathrm{1}}{\mathrm{3}!}+\frac{\mathrm{1}}{\mathrm{4}!}+..} \boldsymbol{\mathrm{dx}} \\ $$$$\:\boldsymbol{\mathrm{n}}=\mathrm{1}\:\rightarrow\frac{\boldsymbol{\mathrm{n}}}{\mathrm{0}!}+\:\frac{\boldsymbol{\mathrm{n}}}{\mathrm{1}!}+\frac{\boldsymbol{\mathrm{n}}}{\mathrm{2}!}+\frac{\boldsymbol{\mathrm{n}}}{\mathrm{3}!}+...=\:\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{n}}} \\ $$$$\:\:\frac{\mathrm{1}}{\mathrm{2}!}+\frac{\mathrm{1}}{\mathrm{3}!}+\frac{\mathrm{1}}{\mathrm{4}!}+..\:\:=\:\boldsymbol{\mathrm{e}}−\mathrm{2} \\ $$$$\:\int\:\boldsymbol{\mathrm{x}}^{\boldsymbol{\mathrm{e}}−\mathrm{2}} \:\boldsymbol{\mathrm{dx}}\:=\:\:\:\frac{\boldsymbol{\mathrm{x}}^{\boldsymbol{\mathrm{e}}−\mathrm{1}} }{\boldsymbol{\mathrm{e}}−\mathrm{1}}\:\:+\boldsymbol{\mathrm{C}} \\ $$$$\:\:\boldsymbol{\mathrm{ABDULAZIZ}}\:\:\:\boldsymbol{\mathrm{ABDUVALIYEV}} \\ $$$$ \\ $$$$ \\ $$

Commented by aliibrahim1 last updated on 21/Jul/21

thx sir amazing

$${thx}\:{sir}\:{amazing} \\ $$

Commented by SEKRET last updated on 21/Jul/21

not at all

$$\:\:\:\:\:\:\:\boldsymbol{\mathrm{not}}\:\boldsymbol{\mathrm{at}}\:\boldsymbol{\mathrm{all}} \\ $$

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