Here is my work, kindly let me know if this is correct:
\begin{align*}\int^t_sxf(x)dx=&\int^{\frac{t-\mu}{\sigma}}_{\frac{s-\mu}{\sigma}}(\sigma z+\mu)\frac{\phi(z)}{\sigma}\sigma dz \\=& \big[\sigma \int z\phi(z)dz+ \mu \int \phi(z) dz\big]^{\frac{t-\mu}{\sigma}}_{\frac{s-\mu}{\sigma}} \\=& \big[-\sigma \phi(z)+ \mu \Phi(z)\big]^{\frac{t-\mu}{\sigma}}_{\frac{s-\mu}{\sigma}} \\=& \sigma [f(s)-f(t)]+ \mu [F(t)-F(s)] \end{align*}
Edited to correct an error I spotted: I should have $f(x)=\phi(z)/\sigma$ instead of $f(x)=\phi(z)$.
Edited to correct a second error I spotted: I should have $\int z \phi(z)dz = -\phi(z)$ instead of $\int z \phi(z)dz = \phi(z)$.
