Question

Integrate the function $\frac{e^{5\log x}-e^{4\log x}}{e^{3\log x}-e^{2\log x}}$

Answer 1

$\frac{e^{5\log x}-e^{4\log x}}{e^{3\log x}-e^{2\log x}}=\frac{e^{4\log x}(e^{\log x}-1)}{e^{2\log x}(e^{\log x}-1)}$
$=e^{2\log x}=e^{log{x}^2}=x^2$
$\therefore \int \frac{e^{5\log x}-e^{4\log x}}{e^{3\log x}-e^{2\log x}}dx=\int x^{2}dx=\frac{x^3}{3}+C$