Question

$\int 5^{5^{5^x}}.5^{5^x}.5^{x}dx$

Answer 1

Let $u=5^{5^x}$

$\therefore x=\frac{\frac{Inu}{In5}}{In5}$

$\therefore dx=\frac{5^{-x-5^x}}{I{n}^{2}5}dx$

$\therefore$ we transform,

$\int 5^{5^{5^x}}{5}^{5^x}{5}^{x}dx=\int 5^4{45}^x\frac{5^{-x-5^x}}{l{n}^{2}5}du$

$=\int \frac{{45}^{4-5}\frac{ln4}{l{n}^{2}5}}{l{n}^{2}5}du$

$=\frac{1}{l{n}^{2}5}\int {45}^{4-5}\frac{ln4}{l{n}^{2}5}du$

$=\frac{1}{l{n}^{2}5}\int 5^{4}du$

$=\frac{1}{l{n}^{2}5}\frac{5^4}{ln5}$

$=\frac{1}{l{n}^{2}5}\frac{5^{5^{5^x}}}{ln5}$

$\therefore \int 5^{5^{5^{5^x}}}{5}^{5^x}{5}^{x}dx=\frac{5^{5^{5^x}}}{l{n}^{3}5}$