证明∫（上标是b,下标是a）f(x)dx=1/2∫（上标是b,下标是a）[f(x)+((ab)/(x^2))*f((ab)/x)]dx

Question

证明∫（上标是b ,下标是a）f(x)dx=1/2∫（上标是b ,下标是a）[f(x)+((ab)
/(x^2))*f(( ab)/x)]dx

Answer 1

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