Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 4 / 29K / a

Write $X=\mu +\sigma Z$, where $Z$ has the standard normal distribution. Apply Gaussian integration by parts to $g(z)=U^{\prime }(\mu +\sigma z)$:Since $X-\mu =\sigma Z$,The polynomial growth assumption and decay of the normal distribution density justify the integration by parts.