Question

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

Answer 1

$$\begin{gathered} \int \left(\frac{e^{6\log x}-e^{5\log x}}{e^{4\log x}-e^{3\log x}}\right)dx \\ =\int \left(\frac{e^{\log x^6}-e^{\log x^5}}{e^{\log x^4}-e^{\log x^3}}\right)dx \\ =\int \left(\frac{x^6-x^5}{x^4-x^3}\right)dx \\ =\int \frac{x^5}{x^3}dx \\ =\int x^{2}dx \\ =\frac{x^3}{3}+C \end{gathered}$$