$\int \frac{2^{x + 1} - 5^{x - 1}}{10^{x}} d x$ is equal to
(where $C$ is constant of integration)

\frac{2 {(\frac{1}{5})}^{x}}{ln (\frac{1}{5})} - \frac{1}{5} \frac{{(\frac{1}{2})}^{x}}{ln (\frac{1}{2})} + C

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\frac{3 {(\frac{1}{5})}^{x}}{ln (\frac{1}{5})} - \frac{1}{5} \frac{{(\frac{1}{2})}^{x}}{ln (\frac{1}{2})} + C

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\frac{4 {(\frac{1}{5})}^{x}}{ln (\frac{1}{5})} - \frac{1}{5} \frac{{(\frac{1}{2})}^{x}}{ln (\frac{1}{2})} + C

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\frac{5 {(\frac{1}{5})}^{x}}{ln (\frac{1}{5})} - \frac{1}{5} \frac{{(\frac{1}{2})}^{x}}{ln (\frac{1}{2})} + C

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Solution

The correct option is A $\frac{2 {(\frac{1}{5})}^{x}}{ln (\frac{1}{5})} - \frac{1}{5} \frac{{(\frac{1}{2})}^{x}}{ln (\frac{1}{2})} + C$
$I = \int \frac{2^{x + 1} - 5^{x - 1}}{10^{x}} d x = \int [2 {(\frac{1}{5})}^{x} - \frac{1}{5} {(\frac{1}{2})}^{x}] d x$
${∵ \int a^{x} = \frac{a^{x}}{ln a}}$
$\Rightarrow I = \frac{2 {(\frac{1}{5})}^{x}}{ln (\frac{1}{5})} - \frac{1}{5} \frac{{(\frac{1}{2})}^{x}}{ln (\frac{1}{2})} + C$

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