证明对任意x≥0arcsin(x-1/x+1)=2arctan√x-∏/2

Question

证明对任意x≥0 arcsin(x-1/x+1)=2arctan√x-∏/2

Answer 1

考虑函数f(x)=arcsin(x-1/x+1)-2arctan√x,x≥0.
对f(x)求导可得f'(x)=0,∴f(x)=C,C为一常数.
又∵f(0)=-π/2,∴f(x)=-π/2.
故arcsin(x-1/x+1)-2arctan√x=-π/2,即arcsin(x-1/x+1)=2arctan√x-π/2.