Let $F=\left(\omega \right)$ be a codim 1 local foliation generated by a germ $\omega$ of the form $\omega =f_1...f_p{\sum }_{i=1}^p{\lambda }_i\frac{d{f}_i}{f_i}$ for some complex numbers ${\lambda }_i$ and germs $f_i$ of holomorphic functions at the origin in ${\mathbf{C}}^n$. We determine, under some conditions, the set of equivalence classes of first order unfoldings and construct explicitly a universal unfolding of $F$. Special cases of this include foliations with holomorphic or meromorphic first integrals. We also show that the unfolding theory for $F$ is equivalent to the unfolding theory for the multiform function...