Question

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

Answer 1

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

$={\int }^{}\frac{2\times 2^x-\frac{5^x}{5}}{2^x\times 5^x}dx$

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

$=-2\times 5^{-x}\log 5+\frac{1}{5}{2}^{-x}\log 2+c$ $\left(\because {\int }^{}{a}^{bx}dx=\frac{a^{bx}\log a}{b}\right)$

$=\frac{-2\log 5}{5^x}+\frac{\log 2}{5\times 2^x}+c$