设f(x)在[0,1]上连续,且定积分∫[0,1]f(x)dx=6,求∫[0,1]dx∫[1,x]f(x)f(y)dy

Question

设f(x)在[0,1] 上连续,且定积分∫[0,1]f(x)dx=6,求∫[0,1]dx∫[1,x]f(x)f(y)dy

Answer 1

设F'(x)=f(x),则
∫[0,1]dx∫[1,x]f(x)f(y)dy
=∫[0,1]f(x)dx∫[1,x]f(y)dy
=∫[0,1]f(x)[F(x)-F(1)]dx
=∫[0,1]f(x)F(x)dx-F(1)∫[0,1]f(x)dx
=F(x)^2/2 |[0,1]-6F(1)
={F(1)-F[0]}{F(1)+F(0)}/2-6F(1)
=3F(1)+3F(0)-6F(1)
=-18