Question

Evaluate: $I=\int \frac{e^x}{(1+e^x)(2+e^x)}$ dx

Answer 1

$\begin{array}{c}{\int }^{}\frac{e^x}{\left(1+e^x\right)\left(x+e^x\right)}dx\\ Let,\\ 1+e^x=t\\ \frac{dt}{dx}=e^x\\ ={\int }^{}\frac{dt}{t\left(t+1\right)}\\ ={\int }^{}\frac{dt}{t^2+t}\\ ={\int }^{}\frac{dt}{{\left(t+\frac{1}{2}\right)}^2-{\left(\frac{1}{2}\right)}^2}\\ =\frac{1}{2\times \frac{1}{2}}\ln \left|\frac{\left(t+\frac{1}{2}\right)-\frac{1}{2}}{\left(t+\frac{1}{2}\right)+\frac{1}{2}}\right|+C\\ =\ln \left|\frac{t}{t+1}\right|+C\\ I=\ln \left|\frac{1+e^x}{2+e^x}\right|+C\end{array}$