A) \[\frac{{{e}^{2}}+2e-3}{2}\]
B) \[\frac{-{{e}^{2}}+2e-3}{2}\]
C) \[\frac{-{{e}^{2}}+2e+1}{2}\]
D) \[\frac{-{{e}^{2}}+2e-1}{2}\]
Correct Answer: B
Solution :
\[y=f(x)\] \[\frac{dy}{dx}=y\] \[\Rightarrow \] \[\frac{dy}{y}=dx\] \[\Rightarrow \] \[y=k\cdot \,{{e}^{x}}\,\And \,f(0)=1\] \[\Rightarrow \] \[k=1\] \[f(x)={{e}^{x}}\] \[\Rightarrow \] \[g(x)={{x}^{2}}-{{e}^{x}}\] \[\int\limits_{0}^{1}{{{e}^{x}}({{x}^{2}}-{{e}^{x}})dx=\left| {{x}^{2}}{{e}^{x}}-2x{{e}^{x}}+2{{e}^{x}}-\frac{{{e}^{2x}}}{2} \right|_{0}^{1}}\]\[=e-2e+2e-\frac{{{e}^{2}}}{2}-2+\frac{1}{2}=\frac{-{{e}^{2}}+2e-3}{2}\]
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