Question

Evaluate $\int$ $\frac{1}{(e^x-1)}dx$

Answer 1

$\int \frac{\frac{1}{e^x}}{1-\frac{1}{e^x}}dx$.

Now denominators is $f\left(x\right)$ & numerator $1$, $f\left(x\right)$ , leave $f^{\prime }\left(x\right)dx$ and treat denominator as 'x' only and performed integration.

$=ln|1-\frac{1}{e^x}|+C$

$=ln\left|\frac{e^{x-1}}{e^x}\right|+C$

$=ln|e^x-1|=ln|e^x|+C$

$=ln|e^x-1|-1+C$.