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Evaluate the following integrals:

Question:

Evaluate the following integrals:

$\int 5^{5^{5^{x}}} 5^{5^{x}} 5^{x} d x$

Solution:

Assume $5^{5^{5^{x}}}=t$

$\Rightarrow \mathrm{d}\left(5^{5^{5^{x}}}\right)=\mathrm{dt}$

$\Rightarrow 5^{5^{x}} \cdot 5^{5^{x}} 5^{x}\left(\log 5^{3}\right) d x=d t$

Substituting $t$ and $d t$

$\Rightarrow 5^{5^{5^{x}}} \cdot 5^{5^{x}} 5^{x} \cdot d x=\frac{d t}{\left(\log 5^{3}\right)}$

$\Rightarrow \int \frac{d t}{\left(\log 5^{2}\right)}$

$\Rightarrow \frac{1}{\left(\log 5^{3}\right)} \int \mathrm{dt}+\mathrm{c}$

$\Rightarrow \frac{t}{\left(\log 5^{3}\right)}+c$

But $t=5^{5^{5^{x}}}$

$\Rightarrow \frac{5^{5^{x^{x}}}}{\left(\log 5^{2}\right)}+c$

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