A) \[\left( \frac{x-1}{x+1} \right){{e}^{x}}+c\]
B) \[{{e}^{x}}\left( \frac{x+1}{x-1} \right)+c\]
C) \[{{e}^{x}}(x+1)(x-1)+c\]
D) None of these
Correct Answer: A
Solution :
\[\int_{{}}^{{}}{\frac{{{e}^{x}}({{x}^{2}}+1)}{{{(x+1)}^{2}}}\,dx}=\int_{{}}^{{}}{\frac{{{e}^{x}}({{x}^{2}}-1+2)}{{{(x+1)}^{2}}}\,dx}\] \[=\int_{{}}^{{}}{{{e}^{x}}\left[ \frac{x-1}{x+1}+\frac{2}{{{(x+1)}^{2}}} \right]}\,dx=\int_{{}}^{{}}{{{e}^{x}}[f(x)+{f}'(x)]\,dx}\] where \[f(x)=\frac{x-1}{x+1}\] and \[{f}'(x)=\frac{2}{{{(x+1)}^{2}}}={{e}^{x}}\left( \frac{x-1}{x+1} \right)+c\].
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