Question

The value of $\int 5^{5^{5^x}}\cdot 5^{5^x}\cdot 5^{x}dx$ is equal to?

• A. $\frac{5^{5^x}}{(ln5{)}^3}+C$
• B. $5^{5^{5^x}}(ln5{)}^3+C$
• C. $\frac{5^{5^{5^x}}}{(ln5{)}^3}+C$
• D. $\frac{5^{5^{5^x}}}{(ln5{)}^2}+C$

Answer 1

Answer: (C)

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

Consider the given integral.

$I=\int 5^{5^{5^x}}\cdot 5^{5^x}\cdot 5^{x}dx$

Let, $u=5^{5^x}\Rightarrow x=\frac{\frac{\ln u}{\ln 5}}{\ln 5}$

$du=5^{x+5^x}{\ln }^2\left(5\right)dx$

$dx=\frac{5^{-x-5^x}}{{\left(\ln 5\right)}^2}du$

Thus, we have,

$I=\int 5^{u}u{5}^x\frac{5^{-x-5^x}}{{\left(\ln 5\right)}^2}du$

$I=\int \frac{u{5}^{u-5\frac{\frac{\ln u}{\ln 5}}{\ln 5}}}{{\left(\ln 5\right)}^2}du$

$I=\frac{1}{{\left(\ln 5\right)}^2}\int 5^{u}du$

$I=\frac{1}{{\left(\ln 5\right)}^2}\frac{5^u}{\ln \left(5\right)}+C$

$I=\frac{1}{{\left(\ln \left(5\right)\right)}^3}+C$

$I=\frac{5^{5^{5^x}}}{{\left(\ln \left(5\right)\right)}^3}+C$

Hence, this is the required result.