A) \[{{e}^{x}}\]
B) \[{{e}^{-x}}\]
C) \[\log x\]
D) \[\frac{{{e}^{{{x}^{2}}}}}{2}\]
E) \[{{e}^{{{x}^{2}}}}\]
Correct Answer: D
Solution :
Let us assume\[f(x)=\frac{{{e}^{{{x}^{2}}}}}{2}\] \[\therefore \]\[\int{x.}\frac{{{e}^{{{x}^{2}}}}}{2}dx=\int{2x\frac{{{e}^{{{x}^{2}}}}}{4}}dx\] Let \[{{x}^{2}}=t\] \[\Rightarrow \] \[2x\,dx=dt\] \[\therefore \] \[\int{\frac{{{e}^{t}}}{4}}dt=\frac{{{e}^{t}}}{4}+c=\frac{1}{2}\left( \frac{{{e}^{{{x}^{2}}}}}{2} \right)+c\] \[\therefore \]\[\int{x}f(x)dx=\frac{1}{2}f(x)+c\]when \[f(x)=\frac{{{e}^{{{x}^{2}}}}}{2}.\]
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