A) \[\log [(1+{{e}^{x}})(2+{{e}^{x}})]+c\]
B) \[\log \left[ \frac{1+{{e}^{x}}}{2+{{e}^{x}}} \right]+c\]
C) \[\log [(1+{{e}^{x}})\sqrt{2+{{e}^{x}}}]+c\]
D) None of these
Correct Answer: B
Solution :
\[\int_{{}}^{{}}{\frac{{{e}^{x}}}{(1+{{e}^{x}})(2+{{e}^{x}})}\,dx}=\int_{{}}^{{}}{\left\{ \frac{{{e}^{x}}}{1+{{e}^{x}}}-\frac{{{e}^{x}}}{2+{{e}^{x}}} \right\}dx}\] Now put \[1+{{e}^{x}}=t\] and \[2+{{e}^{x}}=t,\] then the required integral \[=\log (1+{{e}^{x}})-\log (2+{{e}^{x}})=\log \left( \frac{1+{{e}^{x}}}{2+{{e}^{x}}} \right)+c.\]
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