Question Number 147444 by aliibrahim1 last updated on 20/Jul/21
Answered by SEKRET last updated on 21/Jul/21
$$\int\:\boldsymbol{\mathrm{x}}^{\frac{\mathrm{1}}{\mathrm{2}}} \centerdot\boldsymbol{\mathrm{x}}^{\frac{\mathrm{1}}{\mathrm{2}\centerdot\mathrm{3}}} \centerdot\boldsymbol{\mathrm{x}}^{\frac{\mathrm{1}}{\mathrm{2}\centerdot\mathrm{3}\centerdot\mathrm{4}}} \centerdot….\boldsymbol{\mathrm{dx}}= \\ $$$$\int\:\boldsymbol{\mathrm{x}}^{\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}\centerdot\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{2}\centerdot\mathrm{3}\centerdot\mathrm{4}}+} …\:\mathrm{d}\boldsymbol{\mathrm{x}}=\:\int\:\boldsymbol{\mathrm{x}}^{\frac{\mathrm{1}}{\mathrm{2}!}+\frac{\mathrm{1}}{\mathrm{3}!}+\frac{\mathrm{1}}{\mathrm{4}!}+..} \boldsymbol{\mathrm{dx}} \\ $$$$\:\boldsymbol{\mathrm{n}}=\mathrm{1}\:\rightarrow\frac{\boldsymbol{\mathrm{n}}}{\mathrm{0}!}+\:\frac{\boldsymbol{\mathrm{n}}}{\mathrm{1}!}+\frac{\boldsymbol{\mathrm{n}}}{\mathrm{2}!}+\frac{\boldsymbol{\mathrm{n}}}{\mathrm{3}!}+…=\:\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{n}}} \\ $$$$\:\:\frac{\mathrm{1}}{\mathrm{2}!}+\frac{\mathrm{1}}{\mathrm{3}!}+\frac{\mathrm{1}}{\mathrm{4}!}+..\:\:=\:\boldsymbol{\mathrm{e}}−\mathrm{2} \\ $$$$\:\int\:\boldsymbol{\mathrm{x}}^{\boldsymbol{\mathrm{e}}−\mathrm{2}} \:\boldsymbol{\mathrm{dx}}\:=\:\:\:\frac{\boldsymbol{\mathrm{x}}^{\boldsymbol{\mathrm{e}}−\mathrm{1}} }{\boldsymbol{\mathrm{e}}−\mathrm{1}}\:\:+\boldsymbol{\mathrm{C}} \\ $$$$\:\:\boldsymbol{\mathrm{ABDULAZIZ}}\:\:\:\boldsymbol{\mathrm{ABDUVALIYEV}} \\ $$$$ \\ $$$$ \\ $$
Commented by aliibrahim1 last updated on 21/Jul/21
$${thx}\:{sir}\:{amazing} \\ $$
Commented by SEKRET last updated on 21/Jul/21
$$\:\:\:\:\:\:\:\boldsymbol{\mathrm{not}}\:\boldsymbol{\mathrm{at}}\:\boldsymbol{\mathrm{all}} \\ $$