= Solution
The terminal variable $F=\phi(W_T)$ is bounded and hence <square-integrable>. In the completed natural <Brownian filtration>, the <Brownian martingale representation theorem> says that any square-integrable $\mathcal F_T$-measurable variable admits a representation
$$
F=\mathbb E F+\int_0^T\beta_u\,dW_u
$$
with $\beta$ predictable and $\mathbb E\int_0^T\beta_u^2du<\infty$. Extend $\beta$ by zero after $T$. Thus the requested constant and integrability are
$$
\boxed{c=\mathbb E\phi(W_T),\qquad\mathbb E\int_0^\infty\beta_u^2du=\operatorname{Var}(\phi(W_T)).}
$$
Expectation determines $c$ uniquely. If two integrands give the same representation, the <Itô isometry> gives $\mathbb E\int_0^\infty(\beta_u-\widetilde\beta_u)^2du=0$. Consequently \b[$\beta$ is unique up to $d\mathbb P\,du$-almost everywhere equality], rather than pointwise equality at every time. The corresponding integral martingales are indistinguishable.
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