计算定积分I=∫(0→π)f(sinx)/[f(sinx)+f(cosx)]*dx,其中f(x)为连续函数,且f(sinx)+f(cosx)不等于0

令u=π/2 -x
则x=π/2 -u
原积分=∫(π/2→0) f( sin(π/2 -u) ) / [f( sin(π/2 -u) ) + f( cos(π/2 -u) )] d(π/2 -u)
= - ∫(π/2→0) f(cosu) / [f(cosu) + f(sinu)] du
= ∫(0→π/2) f(cosu) / [f(cosu) + f(sinu)] du
积分变量的字符对积分的结果没有影响,因此
∫(0→π/2)f(sinx)/[f(sinx)+f(cosx)]*dx = ∫(0→π/2)f(cosx)/[f(sinx)+f(cosx)]*dx
则
2 ∫(0→π/2)f(sinx)/[f(sinx)+f(cosx)]*dx
=∫(0→π/2)f(sinx)/[f(sinx)+f(cosx)]*dx + ∫(0→π/2)f(cosx)/[f(sinx)+f(cosx)]*dx
=∫(0→π/2) [f(sinx)+f(cosx)] / [f(sinx)+f(cosx)] dx
=∫(0→π/2) 1 dx
= x|(0→π/2)
= π/2 - 0
= π/2
则原积分 = π/4