Question

Integrate the following functions w.r.t. x.

$\int \frac{e^{5logx}-e^{4logx}}{e^{3logx}-e^{2logx}}dx.$

Answer 1

$\int \frac{e^{5logx}-e^{4logx}}{e^{3logx}-e^{2logx}}dx=\int \frac{e^{log{x}^5}-e^{log{x}^4}}{e^{log{x}^3}-e^{log{x}^2}}dx=\int \frac{x^5-x^4}{x^3-x^2}dx(\because e^{logx}=x)=\int \frac{x^4(x-1)}{x^2(x-1)}dx=\int x^{2}dx=\frac{1}{3}{x}^3+C$