A) \[\int_{\,0}^{\,1}{\,f(x)\,dx}\]
B) \[\int_{\,0.5}^{\,1.5}{\,f(x)\,dx}\]
C) \[2\int_{\,0.5}^{\,1.5}{\,f(x)\,dx}\]
D) 0
Correct Answer: B
Solution :
\[I=\int_{0.5}^{1.5}{xf\,(x)\,dx=\int_{0.5}^{1.5}{(2-x)f(2-x)\,dx}}\], \[\left[ \because \int_{a}^{b}{f(x)dx=\int_{a}^{b}{f(a+b-x)dx}} \right]\] \[=\int_{0.5}^{1.5}{(2-x)f(x)\,dx}=2\int_{0.5}^{1.5}{f(x)\,dx-I}\]Þ\[I=\int_{0.5}^{1.5}{f(x)\,dx}\].
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