Question

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

Answer 1

$\int \frac{e^{5\log x}-e^{4\log x}}{e^{3\log x}-e^{2\log x}}$
$=\int \frac{e^{\log x^5}-e^{\log x^4}}{e^{\log x^3}-e^{\log x^2}}dx$
$[\because e^{b\log a}=a^b]$
$=\int \frac{x^5-x^4}{x^3-x^2}dx$
$=\int \frac{x^2(x^3-x^2)}{x^3-x^2}dx$
$=\int x^{2}dx$
$=\frac{x^3}{3}+C$
Hence,
$\int \frac{e^{5\log x}-e^{4\log x}}{e^{3\log x}-e^{2\log x}}=dx=\frac{x^3}{3}+C$
Where $C$ is constant of integration.