Question

Evaluate :${\int }_0^1\frac{e^{5lnx}-e^{4lnx}}{e^{3lnx}-e^{2lnx}}dx$

• A. $\frac{1}{3}$
• B. 1
• C. 0
• D. $\frac{1}{2}$

Answer 1

The correct option is A $\frac{1}{3}$

${\int }_0^1\frac{e^{5lnx}-e^{4lnx}}{e^{3lnx}-e^{2lnx}}dx={\int }_0^1\frac{x^5-x^4}{x^3-x^2}dx[\because e^{nlnx}=x^n]$
$={\int }_0^1\frac{x^4(x-1)}{x^2(x-1)}dx={\int }_0^1{x}^{2}dx={\left[\frac{x^3}{3}\right]}_0^1=\frac{1}{3}$