Solution
= Solution
Since $B_t=\int_0^t1\,dB_s$, the <Itô isometry> and polarization give
$$
\boxed{\mathbb E\left[B_t\int_0^tf(s)dB_s\right]
=\int_0^t1\cdot f(s)ds
=\int_0^tf(s)ds.}
$$
= Solution
Since $B_t=\int_0^t1\,dB_s$, the <Itô isometry> and polarization give
$$
\boxed{\mathbb E\left[B_t\int_0^tf(s)dB_s\right]
=\int_0^t1\cdot f(s)ds
=\int_0^tf(s)ds.}
$$