A) \[0\]
B) \[\log \left( \frac{e}{1+e} \right)\]
C) \[\log \left( \frac{1}{1+e} \right)\]
D) \[\log (1+e)\]
E) \[1\]
Correct Answer: E
Solution :
Let \[I=\int_{{{e}^{-1}}}^{e}{\frac{dt}{t(t+1)}}\] \[=\int_{{{e}^{-1}}}^{e}{\frac{1}{t}}dt-\int{\frac{1}{(t+1)}}dt\] \[=[\log t-\log (t+1)]_{{{e}^{-1}}}^{e}\] \[=[\log e-\log (e+1)]-[\log {{e}^{-1}}-\log ({{e}^{-1}}+1)]\] \[=\log \left( \frac{e}{e+1} \right)-[-\log e-\log ({{e}^{-1}}+1)]\] \[=\log \left( \frac{e}{e+1} \right)+\left[ \log e\left( \frac{1}{e}+1 \right) \right]\]
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