Question

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

Answer 1

$$\begin{gathered} \int \sqrt{1+e^x}·e^{x}dx \\ \mathrm{Let}1+e^x=t \\ \Rightarrow e^x=\frac{dt}{dx} \\ \Rightarrow e^{x}dx=dt \\ \mathrm{Now},\int \sqrt{1+e^x}·e^{x}dx \\ =\int \sqrt{t}·dt \\ =\frac{t^{{\displaystyle \frac{1}{2}}+1}}{{\displaystyle \frac{1}{2}}+1}+\mathrm{C} \\ =\frac{2}{3}{t}^{\frac{3}{2}}+\mathrm{C} \\ =\frac{2}{3}{\left(1+e^x\right)}^{\frac{3}{2}}+\mathrm{C} \end{gathered}$$