A) \[\frac{{{e}^{x}}}{x+1}+c\]
B) \[\frac{{{e}^{x}}}{{{(x+1)}^{2}}}+c\]
C) \[\frac{{{e}^{x}}}{{{(x+1)}^{3}}}+c\]
D) none of these
Correct Answer: A
Solution :
Let \[I=\int_{{}}^{{}}{\frac{x{{e}^{x}}}{{{(1+x)}^{2}}}}\,dx\] \[=\int_{{}}^{{}}{\frac{{{e}^{x}}(1+x-1)}{{{(1+x)}^{2}}}}\,dx\] \[=\int_{{}}^{{}}{\frac{{{e}^{x}}}{(1+x)}dx-\int_{{}}^{{}}{\frac{{{e}^{x}}}{{{(1+x)}^{2}}}dx}}\] \[=\frac{{{e}^{x}}}{(1+x)}+\int_{{}}^{{}}{\frac{{{e}^{x}}}{{{(1+x)}^{2}}}dx}-\int_{{}}^{{}}{\frac{{{e}^{x}}}{{{(1+x)}^{2}}}dx+c}\] \[=\frac{{{e}^{x}}}{(1+x)}+c\] Note: Integration of \[\int_{{}}^{{}}{{{e}^{x}}(f(x)+f(x))\,dx}\] \[={{e}^{x}}f(x)+c.\]
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