Question

$\int \frac{2^{x+1} - 5^{x-1}}{10^{x}}dx =$

• A. $\frac{2}{\log 5}5^{x} + \frac{1}{5\log2}2^{x} +C$
• B. $\frac{- 2}{\log 5}5^{-x} + \frac{1}{5\log2}2^{-x} +C$
• C. $\frac{1}{\log 5}5^{- x} + \frac{1}{5 \log2}2^{-x} +C$
• D. $none \,of \,these$

Answer 1

Answer: (B)

$\frac{- 2}{\log 5}5^{-x} + \frac{1}{5\log2}2^{-x} +C$

Answer 2

$\int \frac{2^{x+1} - 5^{x - 1}}{10^x} dx$
$\int \frac{2^{x+1}}{\left(2 \times5\right)^{x} } -\frac{5^{x-1}}{\left(2\times 5 \right)^{x}} dx$
$= \int \left(\frac{2}{5^{x} } - \frac{5^{-1}}{2^{x}}\right)dx = \int \left(2\left(5\right)^{-x} -\frac{1}{5} 2^{-x}\right)dx$
$= 2\left(\frac{5^{-x}}{- \log5 }\right) - \frac{1}{5} \left( \frac{2^{-x}}{-\log 2}\right)+C$
$= -\frac{2}{\log 5}5^{-x} + \frac{1}{5\log 2}2^{-x} + C$