The area (in sq. units) of the region bounded by the curves

Question:

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

  1. (1) $\log _{e} 2+\frac{3}{2}$

  2. (2) $\frac{3}{2}$

  3. (3) $\frac{1}{2}$

  4. (4) $\frac{3}{2}-\frac{1}{\log _{e} 2}$


Correct Option: 4,

Solution:

Area $=\int_{0}^{1}\left((x+1)-2^{x}\right) d x$ $\left(\because\right.$ Area $\left.=\int y d x\right)$

$=\left[\frac{x^{2}}{2}+x-\frac{2^{x}}{\ln 2}\right]_{0}^{1}=\left(\frac{1}{2}+1-\frac{2}{\ln 2}\right)-\left(\frac{-1}{\ln 2}\right)=\frac{3}{2}-\frac{1}{\ln 2}$

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