Integration of Brownian Motion with a function of t

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What would be integration of Brownian Motion with $t$ and is there a general formula for $f(t)$ , i.e.

  1. What is $E[\int_{0}^{t} t W_t dt]$? Is it $0$?

  2. Any general rule or result for $E[\int_{0}^{t} f(t) W_t dt]$?