$\int \frac{2^{x+1}-5^{x-1}}{10^x} d x$ is equal to |
$\frac{2}{\ln 5} 5^x+\frac{1}{5 \ln 2} 2^x+c$ $\frac{-2}{\ln 5} 5^{-x}+\frac{1}{5 \ln 2} 2^{-x}+c$ $\frac{1}{2 \ln 5} 5^{-x}-\frac{1}{5 \ln 2} 2^{-x}+c$ none of these |
$\frac{-2}{\ln 5} 5^{-x}+\frac{1}{5 \ln 2} 2^{-x}+c$ |
$\int \frac{2^{x+1}-5^{x-1}}{10^x} d x=\int\left[2\left(\frac{1}{5}\right)^x-\frac{1}{5}\left(\frac{1}{2}\right)^x\right] d x$ $=\frac{2\left(\frac{1}{5}\right)^x}{\ln \frac{1}{5}}-\frac{\frac{1}{5}\left(\frac{1}{2}\right)^x}{\ln \frac{1}{2}}+c$ $=\frac{-2}{\ln 5} 5^{-x}+\frac{1}{5 \ln 2} 2^{-x}+c$ Hence (2) is the correct answer. |