$\int {\frac{{{e^{6\log x}} - {e^{5\log x}}}}{{{e^{4\log x}} - {e^{3\log x}}}}} dx$

}

Let $I = \int {\left( {\frac{{{e^{6\log x}} - {e^{5\log x}}}}{{{e^{4\log x}} - {e^{3\log x}}}}} \right)} dx$

$= \int {\left( {\frac{{{e^{\log {x^6}}} - {e^{\log {x^5}}}}}{{{e^{\log {x^4}}} - {e^{\log {x^3}}}}}} \right)} dx$

[ Using log property : $e^{log_ea }=a ]$

$= \int {\left( {\frac{{{x^6} - {x^5}}}{{{x^4} - {x^3}}}} \right)} dx$

$= \int {\left( {\frac{{{x^3} - {x^2}}}{{x - 1}}} \right)} dx = \int {\frac{{{x^2}(x - 1)}}{{x - 1}}} dx$

$= \int {{x^2}} dx = \frac{{{x^3}}}{3} + C$