Let f: R R be a continuous function. If ₀^ 2 f( 2x) ^3 x , dx = ₀^ 4 f( 2x) x (2 - 2x) , dx , then the value of is

Step-by-step solution

Let I = ₀^ 2 f( 2x) ^3 x , dx Using the property ₀^ 2a g(x) , dx = ₀^a g(x) , dx + _a^ 2a g(x) , dx , we can split the integral: I = ₀^ 4 f( 2x) ^3 x , dx + _ 4 ^ 2 f( 2x) ^3 x , dx In the second integral, substitute x = 2 - t , so dx = -dt . The limits change from 4 2 to 4 0 . _ 4 ^ 2 f( 2x) ^3 x , dx = _ 4 ^0 f( 2( 2 - t)) ^3( 2 - t) (-dt) = ₀^ 4 f( 2t) ^3 t , dt Thus, combining the two parts, we get: I = ₀^ 4 f( 2x) ( ^3 x + ^3 x) , dx Now, apply the property ₀^a g(x) , dx = ₀^a g(a - x) , dx on the interval [0,