Evaluate the following integrals:

Assume $1+e^{x}=t$

$\Rightarrow \mathrm{d}\left(1+\mathrm{e}^{\mathrm{x}}\right)=\mathrm{dt}$

$\Rightarrow \mathrm{e}^{\mathrm{x}} \mathrm{dx}=\mathrm{dt}$

$\therefore$ Substituting $\mathrm{t}$ and dt in given equation we get

$\Rightarrow \int \sqrt{t} \cdot d t$

$\Rightarrow \int t^{1 / 2} \cdot d t$

$\Rightarrow \frac{2 t^{\frac{3}{2}}}{3}+c$

But $1+e^{x}=t$

$\Rightarrow \frac{2\left(1+\mathrm{e}^{\mathrm{x}}\right)^{\mathrm{3} / 2}}{3}+\mathrm{c}$