Question

$\huge{\rm{\pink{From\ 2018\ MIT\ Integration\ BEE\ Qulaifier\ :} }}$
$\displaystyle \rm \int \sqrt{x \sqrt[ 3]{ \rm x \sqrt[4]{ \rm x \sqrt[5]{\rm x \sqrt[6]{ \rm x\ .\ .\ .\ .}}}} }\ dx$

Answer 1

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$\displaystyle \sf \red{\int \sqrt{ x \sqrt[3]{ \sf x \sqrt[4]{\sf x \sqrt[5]{\sf x\ .\ .\ }}}}\ dx}$

$\displaystyle \to \sf \int \sqrt{x}\ . \sqrt{\sqrt[3]{\sf x}}\ .\sqrt{\sqrt[3]{ \sqrt[4]{\sf x}}}\ . \sqrt{\sqrt[3]{\sqrt[4]{\sqrt[5]{\sf x}}}}\ .\ .\ .\ dx$

$\displaystyle \to \sf \int x^{\frac{1}{2}}\ .\ x^{\frac{1}{2}.\frac{1}{3}}\ .\ x^{\frac{1}{2}.\frac{1}{3}.\frac{1}{4}}\ .\ x^{\frac{1}{2}.\frac{1}{3}.\frac{1}{4}.\frac{1}{5}}\ .\ .\ .\ dx$

$\displaystyle \sf \to \int x^{\frac{1}{2}}\ .\ x^{\frac{1}{2.3}}\ .\ x^{\frac{1}{2.3.4}}\ .\ x^{\frac{1}{2.3.4.5}}\ .\ .\ .\ dx$

$\displaystyle \to \sf \int x^{\frac{1}{2}+\frac{1}{2.3}+\frac{1}{2.3.4}+\frac{1}{2.3.4.5}+...}\ dx$

$\displaystyle \to \sf \int x^{\frac{1}{1.2}+\frac{1}{1.2.3}+\frac{1}{1.2.3.4}+\frac{1}{1.2.3.4.5}+...}\ dx$

$\bullet\ \; \pink{\textsf{\textbf{n! = n x (n-1)! }}}$

$\displaystyle \to \sf \int x^{\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\frac{1}{5!}+...}\ dx$

$\bullet\ \; \sf \pink{\textsf{\textbf{e = }} \dfrac{\textsf{\textbf{1}}}{\textsf{\textbf{1}}} \textsf{\textbf{ + }} \dfrac{\textsf{\textbf{1}}}{\textsf{\textbf{1!}}} \textsf{\textbf{ + }} \dfrac{\textsf{\textbf{1}}}{\textsf{\textbf{2!}}} \textsf{\textbf{ + }} \dfrac{\textsf{\textbf{1}}}{\textsf{\textbf{3!}}} \textsf{\textbf{ + }}\ .\ .\ .\ }$

$\displaystyle \to \sf \int x^{e-2}\ dx$

$\displaystyle \blue{\leadsto \sf \dfrac{x^{e-1}}{e-1}\ +c}\ \; \green{\bigstar}$

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