Which of these is equal to $\int e^{x \log 5} e^x dx$, where $C$ is the constant of integration?

$\frac{(5e)^x}{\log 5e} + C$

The correct answer is Option (1) → $\frac{(5e)^x}{\log 5e} + C$

$\int e^{(x \log 5)} e^x dx = \int e^{(\log 5^x)} e^x dx$

$= \int 5^x e^x dx \quad [e^{\log x} = x]$

$= \int (5e)^x dx$

$= \frac{(5e)^x}{\log (5e)} + C$