What is the integration of $\int {\frac{{{e^{6\log x}} – {e^{5\log x}}}}{{{e^{4\log x}} – {e^{3\log x}}}}} dx$~~~~~[NCERT Exemp. Q. 4,Page 163 ]

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$

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