A) \[\log \left( \frac{{{e}^{x}}+1}{{{e}^{x}}+2} \right)+c\]
B) \[\log \left( \frac{{{e}^{x}}+2}{{{e}^{x}}+1} \right)+c\]
C) \[\left( \frac{{{e}^{x}}+1}{{{e}^{x}}+2} \right)+c\]
D) \[\left( \frac{{{e}^{x}}+2}{{{e}^{x}}+1} \right)+c\]
E) \[\frac{1}{2}\log \left( \frac{{{e}^{x}}+1}{{{e}^{x}}+2} \right)+c\]
Correct Answer: A
Solution :
Let\[I=\int{\frac{{{e}^{x}}}{(2+{{e}^{x}})({{e}^{x}}+1)}}dx\] Put \[{{e}^{x}}=t\] \[\Rightarrow \] \[{{e}^{x}}dx=dt\] \[\therefore \] \[I=\int{\frac{1}{(2+t)(1+t)}}dt\] \[=\int{\left[ \left( \frac{1}{1+t} \right)-\left( \frac{1}{2+t} \right) \right]}dt\] \[=\log (1+t)-\log (2+t)+c\] \[=\log \left( \frac{1+{{e}^{x}}}{2+{{e}^{x}}} \right)+c\]
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