Question

$\int 5^{5^{5^x}}.5^{5^x}.5^{x}dx$ is equal to

Answer 1

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

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

$\Rightarrow 5^{5^{5^x}}.ln{5.5}^{5^x}.ln{5.5}^x.In5dx=dt$

$\Rightarrow 5^{5^{5^x}}{.5}^{5^x}{.5}^{x}dx=\frac{dt}{(ln5{)}^3}$

$=\int \frac{dt}{(ln5{)}^3}$

$=\frac{1}{(ln5{)}^3}t+C=\frac{5^{5^{5^x}}}{(ln5{)}^3}+C$