A) \[{{e}^{x}}\left( \frac{x+1}{{{x}^{2}}+x+1} \right)+c\]
B) \[{{e}^{x}}\left( \frac{x-1}{{{x}^{2}}+x+1} \right)+c\]
C) \[{{e}^{x}}\left( \frac{2x-1}{{{x}^{2}}+x+1} \right)+c\]
D) \[{{e}^{x}}\left( \frac{2x+1}{{{x}^{2}}+x+1} \right)+c\]
Correct Answer: B
Solution :
\[\int_{{}}^{{}}{{{e}^{x}}\left( \frac{{{x}^{3}}-1-{{x}^{2}}+2x+2}{{{({{x}^{2}}+x+1)}^{2}}} \right)dx}\] \[=\int_{{}}^{{}}{{{e}^{x}}\left( \frac{x-1}{{{x}^{2}}+x+1}+\frac{-{{x}^{2}}+2x+2}{{{({{x}^{2}}+x+1)}^{2}}} \right)dx}\] \[={{e}^{x}}\left( \frac{x-1}{{{x}^{2}}+x+1} \right)+c\]
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