\( \int\displaystyle\frac{2^{x+1}-5^{x-1}}{10^{x}}\,dx \) is equal to (where \( C \) is an arbitrary constant)
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( -\displaystyle\frac{2}{5^{x}\log_{e}5}+\frac{1}{5\left(2^{x}\log_{e}2\right)}+C \)
Since \( \frac{2^{x}}{10^{x}}=5^{-x} \) and \( \frac{5^{x}}{10^{x}}=2^{-x} \), the integrand splits into two simple terms:
\( \displaystyle \frac{2^{x+1}-5^{x-1}}{10^{x}}=\frac{2\cdot 2^{x}-\frac{1}{5}\cdot 5^{x}}{10^{x}} \)
\( \displaystyle =2\cdot 5^{-x}-\frac{1}{5}\cdot 2^{-x} \)
Using \( \int a^{-x}dx=-\frac{a^{-x}}{\log_{e}a} \):
\( \displaystyle \int\left(2\cdot 5^{-x}-\frac{1}{5}\cdot 2^{-x}\right)dx=-\frac{2}{5^{x}\log_{e}5}+\frac{1}{5\cdot 2^{x}\log_{e}2}+C \)