Evaluate the following integrals: 5^ 5^ 5^ x 5^ 5^ x 5^x dx - Vidyadip

Question

Evaluate the following integrals:
$\int5^{5^{5^{\text{x}}}}5^{5^{\text{x}}}5^\text{x}\text{ dx}$

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Answer

$\int5^{5^{5^{\text{x}}}}5^{5^{\text{x}}}5^\text{x}\text{ dx}$
Let $5^\text{x}=\text{t}$
$\Rightarrow5^\text{x}\log5=\frac{\text{dt}}{\text{dx}}$
$\Rightarrow5^\text{x}\text{dx}=\frac{\text{dt}}{\log5}$
Now, $\int5^{5^{5^{\text{x}}}}5^{5^{\text{x}}}5^\text{x}\text{ dx}$
$=5\int5^{5^{\text{t}}}.5^\text{t}.\frac{\text{dt}}{\log5}$
Again let $5^\text{t}=\text{p}$
$\Rightarrow5^\text{t}\log5=\frac{\text{dp}}{\text{dt}}$
$\Rightarrow5^\text{t}\text{dt}=\frac{\text{dp}}{\log5}$
Again $\int5^{5^\text{t}}.5^\text{x}.\frac{\text{dt}}{\log5}$
$=\int5^\text{p}.\frac{\text{dp}}{(\log5)^2}$
$=\frac{5^\text{p}}{(\log5)^3}+\text{C}$
$=\frac{5^{5^{5^{\text{x}}}}}{(\log5)^3}+\text{C}$

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