If c is any arbitrary constant, then 2^ 2^ 2^x 2^ 2^x 2^x dx is e… - Vidyadip

MCQ

If $c$ is any arbitrary constant, then $\int {{2^{{2^{{2^x}}}}}{2^{{2^x}}}{2^x}dx} $ is equal to

A
$\frac{{{2^{{2^x}}}}}{{{{(\ln 2)}^3}}} + c$
✓
$\frac{{{2^{{2^{{2^x}}}}}}}{{{{(\ln 2)}^3}}} + c$
C
${2^{{2^{{2^x}}}}}{(\ln 2)^3} + c$
D
None of these

✓

Answer

Correct option: B.

$\frac{{{2^{{2^{{2^x}}}}}}}{{{{(\ln 2)}^3}}} + c$

b
(b) Putting ${2^{{2^{{2^x}}}}} = t \Rightarrow {2^{{2^{{2^x}}}}}{2^{{2^x}}}{2^x}{(\log 2)^3}dx = dt,$ we get
$\int_{}^{} {{2^{{2^{{2^x}}}}}{{.2}^{{2^x}}}{{.2}^x}dx} = \frac{1}{{{{(\log 2)}^3}}}\int_{}^{} {1\,.\,dt} = \frac{t}{{{{(\log 2)}^3}}} + c$
$ = \frac{{{2^{{2^{{2^x}}}}}}}{{{{(\log 2)}^3}}} + c$.

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