A) \[\frac{\int{{{2}^{{{2}^{x}}}}}\,dx}{{{(In\,\,2)}^{3}}}+c\]
B) \[\frac{\int{{{2}^{{{2}^{{{2}^{x}}}}}}}\,dx}{{{(In\,\,2)}^{3}}}+c\]
C) \[\int{{{2}^{{{2}^{{{2}^{x}}}}}}}\,\,{{(In\,\,2)}^{3}}+c\]
D) None of the above
Correct Answer: D
Solution :
Let \[I=\int{{{2}^{{{2}^{{{2}^{x}}}}}}}.\,\,\,{{2}^{{{2}^{x}}}}.\,\,{{2}^{x}}dx\] Put \[{{2}^{{{2}^{{{2}^{x}}}}}}=t\] \[\Rightarrow \] \[{{2}^{{{2}^{{{2}^{x}}}}}}{{.2}^{{{2}^{x}}}}{{.2}^{x}}{{(\log \,2)}^{3}}dx=dt\] \[\therefore \] \[I=\int{\frac{dt}{{{(\log \,2)}^{3}}}}\] \[\therefore \] \[I=\frac{t}{{{(\log \,2)}^{3}}}=\frac{{{2}^{{{2}^{{{2}^{x}}}}}}}{{{(\log \,2)}^{3}}}+c\]
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