Question

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

Answer 1

We need to evaluate $\int \frac{1}{e^x-1}dx$

Consider $\frac{1}{e^x-1}$

$\frac{1}{e^x-1}=\frac{e^x}{e^x-1}-1$

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

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

$\therefore \int \frac{1}{e^x-1}dx=ln|e^x-1|-x+c$ where $c$ is integration constant.