Past exam of the year 1 of the computer science course of the University of Oxford / 2013 / Discrete mathematics, probability and continuous mathematics / 5 / a

Differentiate the product $4(x-1{)}^{-2}(x+1{)}^{-1}$ using the product rule and chain rule:Combining terms over a common denominator gives

For a partial fraction decomposition, writeMultiplying by $(x-1{)}^2(x+1)$ gives $4=A(x^2-1)+B(x+1)+C(x-1{)}^2$. Setting $x=1,-1$ gives $B=2,C=1$, and comparing the coefficient of $x^2$ gives $A=-1$. ThusIntegrating each term gives the antiderivativeThe constant of integration can be chosen independently on each interval separated by $-1$ and $1$.