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