Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2006 / Paper 1 / 9 / b / ii / Solution

For the arctangent, put $h=x-1$. Its derivative is $(d/dh)\arctan (1+h)=1/(2+2h+h^2)=\frac{1}{2}-\frac{h}{2}+\frac{h^2}{4}+O(h^4)$. Integrating and using $\arctan 1=\pi /4$ givesThe constant counts as the first nonzero term. The quartic coefficient vanishes; the next nonzero term is $-h^5/40$.