question_answer

Consider the area bounded by \[f(x)={{e}^{-x}}+3,\] \[g(x)=\log (x+2),\] \[x=-1\] and y-axis. The areas divided by the line passing through the points of intersection of these curves with the coordinate axes are in the ratio

A) \[(3+2\,\,ln\,2):e\]     

B) \[(3+2\,\,ln\,2):e\]

C) \[e:(3+2\,\,ln\,2)\]

D) \[e:3-2\log 2\]

Correct Answer: D

Solution :

[d] Total area \[{{A}_{1}}+{{A}_{2}}=\int\limits_{-1}^{0}{[{{e}^{-x}}+3-\log (x+2)]dx}\] \[=\left[ \frac{{{e}^{-x}}}{-1}+3x \right]_{-1}^{0}-\int\limits_{1}^{2}{\log \,\,tdt}\] \[=-1-(-e-3)-\left[ t\log t-t \right]_{1}^{2}\] \[=e+2-2[\log \,2-1]+(-1)\] \[=e+3-2\log 2\] \[{{A}_{1}}=2-\int\limits_{-1}^{0}{\log \,(x+2)dx}\] \[=2-[2\log 2-1]=3-2\log 2\] \[\therefore \,\,{{A}_{2}}=e+3-2\log2 -3+2\log 2=e\] \[{{A}_{2}}:{{A}_{1}}=e:3-2\log 2\]