$\int e^{(x\log 5)} e^x\, dx$, is:

Where C is the constant of integration.

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

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

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

Combine exponents:

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

So the integrand becomes:

$5^{x}e^{x} = (5e)^{x}$

Thus:

$\displaystyle \int (5e)^{x}\,dx$

Integral of $a^{x}$ is $\frac{a^{x}}{\ln a}$, so:

$\displaystyle \int (5e)^{x}\,dx = \frac{(5e)^{x}}{\ln(5e)} + C$

Final Answer: $\displaystyle \frac{(5e)^{x}}{\ln(5e)} + C$