Question

Find: $\int \frac{e^{x}dx}{(e^x-1{)}^2(e^x+2)}$

Answer 1

Let us consider $e^x-1=t$

We get $e^{x}dx=dt$

So we have $\int \frac{e^{x}dx}{{(e^x-1)}^2(e^x+2)}=\int \frac{dt}{t^2(t+3)}=\frac{1}{9}\int \frac{1}{t+3}dt+\frac{1}{9}\int \frac{-t+3}{t^2}dt$

$=\frac{\ln (t+3)}{9}+\frac{1}{9}(-\ln t-\frac{3}{t})=\frac{\ln (t+3)-\ln t}{9}-\frac{1}{3t}$