Question

Integration of e^5 log x - e^4logx/(e^3logx-e^2logx)

Answer 1

Simply use the property of log that is of e^5logx will be equal to e^logx^5 will be equal to x^5

Answer 2

Answer:

$I=\frac{x^3}{3}+C$

Explanation:

We are given integral

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

First we use log property and simplify the integral.

Log exponent property:

$a^{x\log_ay}=y^x$

Integral change to

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

Now we factor numerator and denominator and we get

$I=\int \frac{x^{4}(x-1)}{x^{2}(x-1)}dx$

Cancel like factor from numerator and denominator

$I=\int x^2 dx$

$I=\frac{x^3}{3}+C$

where, C is integral constant.