$\int \frac{2^{x+1}-5^{x-1}}{10^x} d x$ is equal to

$\frac{-2}{\ln 5} 5^{-x}+\frac{1}{5 \ln 2} 2^{-x}+c$

$\int \frac{2^{x+1}-5^{x-1}}{10^x} d x=\int\left[2\left(\frac{1}{5}\right)^x-\frac{1}{5}\left(\frac{1}{2}\right)^x\right] d x$

$=\frac{2\left(\frac{1}{5}\right)^x}{\ln \frac{1}{5}}-\frac{\frac{1}{5}\left(\frac{1}{2}\right)^x}{\ln \frac{1}{2}}+c$

$=\frac{-2}{\ln 5} 5^{-x}+\frac{1}{5 \ln 2} 2^{-x}+c$

Hence (2) is the correct answer.