Question

$\int \frac{e^x+1}{e^x+x}dx$

Answer 1

$$\begin{gathered} \mathrm{Let}I=\int \frac{e^x+1}{e^x+x}dx \\ \mathrm{Putting}{e}^x+x=t \\ \Rightarrow e^x+1=\frac{dt}{dx} \\ \Rightarrow \left(e^x+1\right)dx=dt \\ \therefore I=\int \frac{1}{t}dt \\ =\ln \left|t\right|+\mathrm{C} \\ =\ln \left|e^x+x\right|+\mathrm{C}\left[\because t={\mathrm{e}}^x+x\right] \end{gathered}$$