A) \[\frac{1}{2}{{\left\{ f(x) \right\}}^{2}}\]
B) \[{{\left\{ f(x) \right\}}^{3}}\]
C) \[\frac{{{\left\{ f(x) \right\}}^{3}}}{3}\]
D) \[{{\left\{ f(x) \right\}}^{2}}\]
Correct Answer: A
Solution :
Since, \[\int{f(x)\,dx}=f(x)\] \[\therefore \] \[\frac{d}{dx}f(x)=f(x)\] \[\left[ \because \,f(x)=\int{\frac{d}{dx}}f(x)dx \right]\] Now, \[\int{{{\{f(x)\}}^{2}}dx}=\int{f(x)\cdot f(x)dx}\] \[=f(x)\int{f(x)dx}-\int{\left[ \frac{d}{dx}f(x)\int{f(x)dx} \right]}\,dx\] (integrating by parts) \[=f(x)f(x)-\int{f(x)f(x)\,dx}\] \[\Rightarrow \] \[2\int{{{\{f(x)\}}^{2}}dx={{\{f(x)\}}^{2}}}\] \[\Rightarrow \] \[\int{{{\{f(x)\}}^{2}}dx}=\frac{1}{2}{{\{f(x)\}}^{2}}\] Alternate \[\int{f(x)\,dx}=f(x)\] \[\left[ \because \int{\frac{d}{dx}f(x)\,dx}=f(x) \right]\] \[f(x)=\frac{d}{dx}f(x)\] \[\Rightarrow \] \[\int{dx}=\int{\frac{d\{f(x)\}}{f(x)}},\]on integrating \[\log c+x=\log f(x)\] \[f(x)=c\,{{e}^{x}}\] \[{{\{f(x)\}}^{2}}={{c}^{2}}{{e}^{2x}}\] On integrating, \[\int{{{\{f(x)\}}^{2}}dx=\frac{{{c}^{2}}}{2}{{e}^{2x}},}\] [From Eq. (i)] \[\int{{{\{f(x)\}}^{2}}}dx=\frac{1}{2}{{\{c{{e}^{x}}\}}^{2}}=\frac{1}{2}{{\{f(x)\}}^{2}}\]
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