Does a stochastic integral depend on the index of the variable of the integration?

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Is true that $\int_0^t f(u)dW_u$ is just $f(t)W_t$ where $W_u$ is Wiener process?
I know that $\int_0^t const \text{ }dW_u = const \cdot W_t$ but this is not the case as $f$ depends on the variable of the integration.