Explanation

(c)
$\int_{}^{}\frac{e^{5\log_{e}x} - e^{4\log_{e}x}}{e^{3\log_{e}x} - e^{2\log_{e}x}}\ dx = \int_{}^{}\frac{e^{\log_{e}x^{5}} - e^{\log_{e}x^{4}}}{e^{\log_{e}x^{3}} - e^{\log_{e}x^{2}}}\ dx$
$= \int_{}^{}{\frac{x^{5} - x^{4}}{x^{3} - x^{2}}dx = \int_{}^{}\frac{x^{4}(x - 1)}{x^{2}(x - 1)}}\ dx$
$= \int_{}^{}x^{2}\ dx = \frac{x^{3}}{3} + c$