_ ^ e^x (1 + e^x )(2 + e^x ) dx = - Vidyadip

MCQ

$\int_{}^{} {\frac{{{e^x}}}{{(1 + {e^x})(2 + {e^x})}}dx = } $

A
$\log [(1 + {e^x})(2 + {e^x})] + c$
✓
$\log \left[ {\frac{{1 + {e^x}}}{{2 + {e^x}}}} \right] + c$
C
$\log [(1 + {e^x})\sqrt {2 + {e^x}} ] + c$
D
None of these

✓

Answer

Correct option: B.

$\log \left[ {\frac{{1 + {e^x}}}{{2 + {e^x}}}} \right] + c$

b
(b)$\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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