Past exam of the options II for mathematics and philosophy course of the University of Oxford / 2013 / D3 / b / Solution

For $x>0$, the geometric series givesEvery summand is nonnegative and has integral $(n+\alpha {)}^{-2}$ by part (a). This numerical series converges by comparison with ${\sum }_{n\ge 1}{n}^{-2}$. The series-integration rule proved above therefore yieldsThe value at the single endpoint zero does not affect this Lebesgue integral.