= Solution
Let $K$ be the function field of the <integral scheme> $X$ and choose a nonzero <rational section of a line bundle> $s\in\mathcal L_\eta$. Choose an open trivializing cover $U_i$ with generators $e_i$ for the <invertible sheaf> $\mathcal L$. On each nonempty $U_i$, write $s=a_ie_i$ with $a_i\in K^*$. If $e_i=u_{ij}e_j$ on an overlap, then $a_j=a_iu_{ij}$, so $a_i/a_j=u_{ij}^{-1}$ is a regular unit. Consequently the local rational functions $a_i$ define a <Cartier divisor> $D$.
The <divisor line bundle> $\mathcal O_X(D)$ restricts to $a_i^{-1}\mathcal O_{U_i}$. Define its local isomorphism from $\mathcal L$ by $e_i\mapsto a_i^{-1}$. On an overlap,
$$
u_{ij}a_j^{-1}=a_i^{-1},
$$
so the maps glue. Hence
$$
\boxed{\mathcal L\cong\mathcal O_X(D).}
$$
The rational section maps to the rational function $1$, which also fixes the sign convention for $D$. Choosing a different nonzero rational section changes $D$ by a <principal Cartier divisor>, so its <linear equivalence of Cartier divisors> class is unchanged. No projectivity or Noetherian hypothesis is required.
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