设f（x）在【0,1】上连续且∫(0,1)f(x)dx=A,证明∫(0,1)dx∫(x,1)f(x)f(y)dy=A∧2/2,谢谢!

题目

设f（x）在【0,1】上连续且∫(0,1)f(x)dx=A,证明∫(0,1)dx∫(x,1)f(x)f(y)dy=A∧2/2,谢谢!

答案

设∫f(x)dx=F(x),则F(0)=0,F(1)=A,
∫[∫f(x)f(y)dy]dx
=∫f(x)[∫dF(y)] dx
=∫[A-F(x)]dF(x )
=A∫f(x)dx -(A^2)/2
=(A^2)/2