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It was shown in another question that \begin{align} \int^T_0 f(t) Z_t dt &= Z_T \int^T_0 f(u) du - \int^T_0 \Bigl( \int^s_0 f(u) du \Bigr) dZ_s\\ &= Z_T \cdot F(T) - \int^T_0 F(s) \cdot dZ_s \end{align} where I've defined the integral $F(T) = \int^T_0 f(t) dt$, and note this is just another deterministic function of time. Both of the two terms in the ...