A) \[\frac{{{e}^{-x}}}{1+x}+c\]
B) \[-\frac{{{e}^{-x}}}{1+x}+c\]
C) \[\frac{{{e}^{x}}}{1+x}+c\]
D) \[-\frac{{{e}^{x}}}{1+x}+c\]
Correct Answer: C
Solution :
\[\int_{{}}^{{}}{\frac{x{{e}^{x}}}{{{(1+x)}^{2}}}\,dx=\int_{{}}^{{}}{\frac{(x+1-1)}{{{(1+x)}^{2}}}{{e}^{x}}dx}}\] \[=\int_{{}}^{{}}{{{e}^{x}}\left( \frac{1}{1+x}-\frac{1}{{{(1+x)}^{2}}} \right)\,dx}=\frac{{{e}^{x}}}{1+x}+c\].
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