$\int e^{\log _5 x} d x$
Answer & explanation
Correct answer: option 2
$e^{\log _5 x}=x^{\log _5 e}$ (by property of exponential function)
$\int e^{\log _5 x} d x=\int x^{\log _5 e} d x=\frac{x^{\log _5 5 e}+1}{\log _5 5 e+1}=\frac{x^{\log _5 5 e}}{\log _5 5 e}$
Hence (2) is the correct answer.