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