A) \[a\int_{0}^{a}{f(x)dx}\]
B) \[\frac{a}{2}\int_{0}^{a}{f(x)dx}\]
C) \[\int_{0}^{a}{f(x)dx}\]
D) None of these
Correct Answer: B
Solution :
\[f(a-x)=f(x)\] Let \[I=\int_{0}^{a}{x\,f(x)}dx\] ...(i) \[\Rightarrow \] \[I=\int_{0}^{a}{(a-x)f(a-x)dx}\] \[\Rightarrow \] \[I=\int_{0}^{a}{(a-x)f(x)dx}\] ...(ii) On adding Eqs. (i) and (ii) \[2I=\int_{0}^{a}{xf(x)dx+\int_{0}^{a}{(a-x)f(x)}dx}\] \[=\int_{0}^{a}{xf(x)dx+\int_{0}^{a}{a\,f(x)}dx}-\int_{0}^{a}{xf(x)}dx\] \[\Rightarrow \] \[2I=a\int_{0}^{a}{f(x)dx}\] \[\Rightarrow \] \[I=\frac{a}{2}\int_{0}^{a}{f(x)dx}\]
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