Find f(x) if f(1)=0 and the tangent line at (x,f(x)) has a slope of x/((x^2)+1)

Answer 1

so, you have
df/dx = x/(x^2+1)
f(x) = 1/2 ln(x^2+1) + c
since f(1) = 0,
1/2 ln2 + c = 0
f(x) = 1/2 ln(x^2+1) - 1/2 ln2