A) \[a\int_{0}^{a}{f(x)}dx\]
B) \[\frac{a}{2}\int_{0}^{a}{x\,f(x)\,}dx\]
C) \[\frac{{{a}^{2}}}{2}\int_{0}^{a}{f(x)\,}dx\]
D) None of these
Correct Answer: B
Solution :
Given,\[f(a-x)=f(x),\]let \[I=\int_{0}^{a}{x}\,f(x)dx\] Put \[x=a-y\] \[\Rightarrow \] \[dx=-dy\] when \[x=0,\text{ }y=a\] \[x=a,\text{ }y=0\] \[\therefore \] \[I=\int_{a}^{0}{(a-y)f(a-y)(-dy)}\] \[=\int_{a}^{0}{(a-x)f(a-x)dx}\] \[=\int_{a}^{0}{(a-x)f(x)dx}\] [given] \[=\int_{a}^{0}{af(x)dx}-\int_{0}^{a}{x\,f(x)}\,dx\] \[\Rightarrow \] \[I=\int_{0}^{a}{af(x)}dx=I\] \[\Rightarrow \] \[2I=\int_{0}^{a}{af(x)}\,dx\] \[\Rightarrow \] \[I=\frac{a}{2}\int_{0}^{a}{f(x)}\,dx\]
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