问题描述: 设f(x)∈C[0,1],证明∫(π,0)*x*f(sinx)dx =π/2*∫(π,0)*f(sinx)dx 1个回答 分类:数学 2014-11-03 问题解答: 我来补答 设x = π - y,dx = - dy当x = 0,y = π当x = π,y = 0∫(0→π) xf(sinx) dx = - ∫(π→0) (π - y)f(sin(π - y)) dy= π∫(0→π) f(siny) dy - ∫(0→π) yf(siny) dy= π∫(0→π) f(sinx) dx - ∫(0→π) xf(sinx) dx,这里的y是假变量2∫(0→π) xf(sinx) dx = π∫(0→π) f(sinx) dx,重复,移项∴∫(0→π) xf(sinx) dx = (π/2)∫(0→π) f(sinx) dx 展开全文阅读