The area (in sq.units) of the region bounded by the curves $y = 2^x$ and$y = \left|\mathrm{x}+1\right|$ in the first
quadrant is

$\mathrm{y}=2^{\mathrm{x}},\mathrm{y}=|\mathrm{x}+1|=\left\{\begin{array}{l}\mathrm{x}+1,\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{x}\ge -1\\ \mathrm{x}-1,\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{x}<-1\end{array}\right.$
Required area $={\int }_0^1 \left((\mathrm{x}+1)-2^{\mathrm{x}}\right)\mathrm{dx}$
$={\left(\frac{{\mathrm{x}}^2}{2}+\mathrm{x}-\frac{2^{\mathrm{x}}}{\log 2}\right)}_0^1=\frac{1}{2}+1-\frac{2}{\log 2}+\frac{1}{{\log }_{\mathrm{e}}^2}$
$\frac{3}{2}-\frac{1}{{\log }_{\mathrm{e}}2}$ sq units