Question

$\int \left(\frac{e^{5\log x}-e^{4\log x}}{e^{3\log x}-e^{2\log x}}\right)dx=$

• A. $x+c$
• B. $3x^2+c$
• C. $\frac{x^3}{3}+c$
• D. $\frac{x^3}{2}+c$

Answer 1

Answer: (C)

$\frac{x^3}{3}+c$

Consider $I=\int \frac{e^{5\log x}-e^{4\log x}}{e^{3\log x}-e^{2\log x}}dx$

We know, $e^{5\log x}=x^5$

$I=\int \frac{x^5-x^4}{x^3-x^2}dx=\int x^2\frac{(x^3-x^2)}{(x^3-x^2)}dx$

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

$\therefore \int \left(\frac{e^{5\log x}-e^{4\log x}}{e^{3\log x}-e^{2\log x}}\right)dx=\frac{x^3}{3}+c$