證明：由于sinx,cosx是連續(xù)函數(shù),而由已知f(u,v)在區(qū)域D=上連續(xù),所以復(fù)合函數(shù)f(sinx,cosx)和f(cosx,sinx)是在0≤x≤π/2是連續(xù)的,因此在0≤x≤π/2上f(sinx,cosx)和f(cosx,sinx)積分都存在.做積分變換y=π/2-x有
∫(π/2)(0)f(sinx,cosx)dx=-∫(0)(π/2)f(sin(π/2-y),cos(π/2-y))dy=∫(π/2)(0)f(cosy,siny)dy=∫(π/2)(0)f(cosx,sinx)dx
證畢.
另外,曲線(xiàn)y=1/x+ln(1+e^x)有兩條漸近線(xiàn)x=0和y=0.