If $$f\left( x \right)$$  and $$\phi \left( x \right)$$  are continuous functions on the interval $$\left[ {0,\,4} \right]$$  satisfying $$f\left( x \right) = f\left( {4 - x} \right),\,\phi \left( x \right) + \phi \left( {4 - x} \right) = 3$$        and $$\int\limits_0^4 {f\left( x \right)dx = 2} ,$$   then $$\int\limits_0^4 {f\left( x \right)\phi \left( x \right)dx} = ?$$

Solution :
$$\eqalign{ & {\text{Let }}I = \int\limits_0^4 {f\left( x \right)\phi \left( x \right)dx} \cr & \Rightarrow I = \int\limits_0^4 {f\left( {4 - x} \right)\phi \left( {4 - x} \right)dx} \cr & \Rightarrow I = \int\limits_0^4 {f\left( x \right).\left( {3 - \phi \left( x \right)} \right)dx} \cr & \left[ {\because \,f\left( x \right) = f\left( {4 - x} \right){\text{ and }}\phi \left( x \right) + \phi \left( {4 - x} \right) = 3} \right] \cr & \Rightarrow I = 3\int\limits_0^4 {f\left( x \right)dx - I} \cr & \Rightarrow 2I = 3.2 \cr & \therefore I = 3 \cr} $$