设f（x）在【0,1】上连续.证明∫（π/2~0)f(cosx)dx=∫（π/2~0)f（sinx）dx

来源：学生作业帮助网 编辑：作业帮 时间：2024/08/15 03:04:11
设f（x）在【0,1】上连续.证明∫（π/2~0)f(cosx)dx=∫（π/2~0)f（sinx）dx
x){n_=t>q zk^o|6Qj Fui)@fqfȨ "}_`gCc?ٽD$[cf] kAHBS,ѥc @SR3b.I(hF 1n6`ӎ!A|mOv,yoY-59iÞw74!XE

设f（x）在【0,1】上连续.证明∫（π/2~0)f(cosx)dx=∫（π/2~0)f（sinx）dx
设f（x）在【0,1】上连续.证明∫（π/2~0)f(cosx)dx=∫（π/2~0)f（sinx）dx

设f（x）在【0,1】上连续.证明∫（π/2~0)f(cosx)dx=∫（π/2~0)f（sinx）dx
令 y=π/2-x,则x=π/2-y
∫（π/2~0)f(cosx)dx=∫（0~π/2) f(cos(π/2-y))d(π/2-y)
=∫（0~π/2) -f(siny)dy
=-∫（0~π/2) f(siny)dy
=∫（π/2~0)f(siny)dy
=∫（π/2~0)f(sinx)dx

分别证明等号两边的极限值相等。