高数函数已知f(x+1/x)=ln|x|-1/2ln(x^4+1),求f(x).

Question

高数函数
已知f(x+1/x)=ln|x|-1/2ln(x^4+1),求f(x).

Answer 1

f(x+1/x) = ln|x|-1/2ln(x^4+1)
= ln|x|-ln[(x^4+1)^(1/2)]
= 1/2*ln[x^2/(x^4+1)]
令t=x+1/x=(x^2+1)/x
则有t^2=(x^4+1)/x^2+2
即x^2/(x^4+1)=1/(t^2-2)
所以f(x+1/x)=f(t)=1/2*ln[1/(t^2-2)]=-1/2ln(t^2-2)
即f(x)=-1/2ln(x^2-2)