Solution
= Solution
The <martingale product identity> and the Itô isometry for cross terms give
$$
\mathbb E\left[B_t\int_0^te^{B_s}\,dB_s\right]
=\mathbb E\int_0^te^{B_s}\,ds.
$$
Since $B_s\sim N(0,s)$, its <moment-generating function> gives $\mathbb Ee^{B_s}=e^{s/2}$. Therefore the answer is
$$
\int_0^te^{s/2}\,ds=2(e^{t/2}-1).
$$