第1个回答 2020-02-28
1/2∫(b,a)[f(x)+((ab)/(x^2))*f((ab)/x)]dx
=1/2∫(b,a)f(x)dx+1/2∫(b,a)[((ab)/(x^2))*f((ab)/x)]dx
前面的f(x)dx暂不考虑,
后面部分,令y=ab/x,则x=ab/y
带入1/2∫(b,a)[((ab)/(x^2))*f((ab)/x)]dx得:
1/2∫(b,a)[y^2/(ab)f(y)]d(ab/y)
因为d(ab/y)=ab/y^2dy
所以1/2∫(b,a)[y^2/(ab)f(y)]d(ab/y)=1/2∫(b ,a)f(y)d(y)
所以1/2∫(b,a)[f(x)+((ab)/(x^2))*f((ab)/x)]dx=∫(b,a)f(x)dx