Passage: Let f(x)= x and g(x)-f(x)=f(4-x) . Question: What is ₀^4 f(4-x) g(4-x) ,dx equal to ?

Step-by-step solution

Given g(x) - f(x) = f(4-x) g(x) = f(x) + f(4-x) Replacing x with 4-x , we get: g(4-x) = f(4-x) + f(x) = g(x) Let I = ₀^4 f(4-x) g(4-x) ,dx Using the property ₀^a h(x) ,dx = ₀^a h(a-x) ,dx , we have: I = ₀^4 f(x) g(x) ,dx Adding the two expressions for I : 2I = ₀^4 f(4-x) g(4-x) ,dx + ₀^4 f(x) g(x) ,dx Since g(4-x) = g(x) , this becomes: 2I = ₀^4 f(4-x) + f(x) g(x) ,dx Substituting f(x) + f(4-x) = g(x) : 2I = ₀^4 g(x) g(x) ,dx = ₀^4 1 ,dx 2I = [x]₀^4 = 4 I = 2 Answer: 2