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Question
\[\int 5^{5^{5^x}} 5^{5^x} 5^x dx\]
Sum
Solution
\[\int 5^{5^{5^x}} \cdot 5^{5^x} \cdot 5^x dx\]
\[\text{Let 5}^x = t\]
\[ \Rightarrow 5^x \log 5 = \frac{dt}{dx}\]
\[ \Rightarrow 5^x dx = \frac{dt}{\log 5}\]
\[Now, \int 5^{5^{5^x}} \cdot 5^{5^x} \cdot 5^x dx\]
\[ = \int 5^{5^t} \cdot 5^t \cdot \frac{dt}{\log 5}\]
\[\text{Again let 5}^t = p\]
\[ \Rightarrow 5^t \log 5 = \frac{dp}{dt}\]
\[ \Rightarrow 5^t dt = \frac{dp}{\log 5}\]
\[Again \int 5^{5^t} \cdot 5^t \cdot \frac{dt}{\log 5}\]
\[ = \int 5^p \cdot \frac{dp}{\left( \log 5 \right)^2}\]
\[ = \frac{5^p}{\left( \log 5 \right)^3} + C\]
\[ = \frac{5^{5^{5^x}}}{\left( \log 5 \right)^3} + C\]
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