Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 1 / 31C / a / Solution

Repeated integration by parts would require derivatives of $f$ and finite endpoint values of those derivatives. The hypothesis gives only continuity away from zero, and even the leading behavior $x^{\alpha }$ can diverge at zero or have singular derivatives when $\alpha$ is nonintegral. Thus those integrations need not be defined. The appropriate method integrates the asymptotic powers themselves, rather than differentiating an insufficiently regular function.