设f(x)∈C[0,1],证明∫（π,0）*x*f(sinx)dx =π/2*∫（π,0）*f(sinx)dx_数学解答

设f(x)∈C[0,1],证明∫（π,0）*x*f(sinx)dx =π/2*∫（π,0）*f(sinx)dx

人气：295 ℃　时间：2019-10-03 08:45:48

解答

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(s...