A) \[2\int\limits_{0}^{a}{f(x)dx}\]
B) \[4\int\limits_{0}^{a}{f(x)dx}\]
C) \[-3\int\limits_{0}^{a}{f(x)dx}\]
D) \[\int\limits_{0}^{a}{f(x)dx}\]
Correct Answer: A
Solution :
Here,\[f(x)=f(a-x)\]and\[g(x)+g(a-x)=4\] Let\[I=\int\limits_{0}^{a}{f(x)g(x)dx}=\int\limits_{0}^{a}{f}(a-x)g(a-x)dx\] \[=\int\limits_{0}^{a}{f}(x)(4-g(x))dx\] \[=4\int\limits_{0}^{a}{f}(x)dx-\int\limits_{0}^{a}{f}(x)g(x)dx\] \[=4\int\limits_{0}^{a}{f}(x)dx-I\Rightarrow I=2\int\limits_{0}^{a}{f}(x)dx\]
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