作业帮设f(x)∈C[0,1],证明∫（π,0）*x*f(sinx)dx =π/2*∫（π,0）*f(sinx)dx
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![设f(x)∈C[0,1],证明∫（π,0）*x*f(sinx)dx =π/2*∫（π,0）*f(sinx)dx](/uploads/image/z/1699231-31-1.jpg?t=%E8%AE%BEf%28x%29%E2%88%88C%5B0%2C1%5D%2C%E8%AF%81%E6%98%8E%E2%88%AB%EF%BC%88%CF%80%2C0%EF%BC%89%2Ax%2Af%28sinx%29dx+%3D%CF%80%2F2%2A%E2%88%AB%EF%BC%88%CF%80%2C0%EF%BC%89%2Af%28sinx%29dx)
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设f(x)∈C[0,1],证明∫（π,0）*x*f(sinx)dx =π/2*∫（π,0）*f(sinx)dx
设f(x)∈C[0,1],证明∫（π,0）*x*f(sinx)dx =π/2*∫（π,0）*f(sinx)dx
设f(x)∈C[0,1],证明∫（π,0）*x*f(sinx)dx =π/2*∫（π,0）*f(sinx)dx
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