$\int e^{(x\log 5)} e^x\, dx$, is:
Where C is the constant of integration.
Answer & explanation
Correct answer: option 1
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: $\displaystyle \frac{(5e)^{x}}{\ln(5e)} + C$