Question

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

• A. $\frac{2}{\log 5}{5}^x+\frac{1}{5\log 2}{2}^x+C$
• B. $\frac{-2}{\log 5}{5}^{-x}+\frac{1}{5\log 2}{2}^{-x}+C$
• C. $\frac{1}{\log 5}{5}^{-x}+\frac{1}{5\log 2}{2}^{-x}+C$
• D. $noneofthese$

Answer 1

Answer: (B)

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