Application of Itô's Lemma

Question

I am confused as to how to begin solving the following question:

Let $B$ be a Brownian Motion. Ust Itô's Lemma to show that $\int_{0}^{t}B_u du = \int_{0}^{t} (t-u)dB_u$.

I tried taking the derivative of both sides in order to get some hint how to proceed, but all I found was that the derivative of right hand side was equal to 0 i.e. $B_tdt = 0$.

Answer 1

$$d(tB_t) = B_t dt + tdB_t$$ hence $$ tB_t = \int_0^t B_udu +\int_0^t udB_u$$ Note that $$ tB_t= \int_0^t tdB_u$$ and so $$ \int_0^t B_udu = \int_0^t(t-u)dB_u $$