A) \[\frac{{{5}^{{{5}^{x}}}}}{{{(log5)}^{3}}}+c\]
B) \[{{5}^{{{5}^{{{5}^{x}}}}}}{{(log5)}^{3}}+c\]
C) \[\frac{{{5}^{{{5}^{{{5}^{x}}}}}}}{{{(log5)}^{3}}}+c\]
D) none of these
Correct Answer: C
Solution :
Let \[I=\int_{{}}^{{}}{{{5}^{{{5}^{{{5}^{x}}}}}}{{.5}^{{{5}^{x}}}}{{.5}^{x}}dx}\] Put \[{{5}^{{{5}^{{{5}^{x}}}}}}=t\]\[\Rightarrow \]\[{{5}^{{{5}^{{{5}^{x}}}}}}{{.5}^{{{5}^{x}}}}{{.5}^{x}}{{(log5)}^{3}}dx=dt\] \[\therefore \] \[I=\frac{1}{{{(log5)}^{3}}}\int_{{}}^{{}}{\frac{t\,dt}{t}}\] \[=\frac{t}{{{(log5)}^{3}}}+c\] \[=\frac{{{5}^{{{5}^{5}}^{x}}}}{{{(log5)}^{3}}}+c\]
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