Mixed moment between a Brownian motion in absolute value and another Brownian motion

Question

I have this given problem, how can I calculate this mixed moment?

$$E[|B(t)|* B(1-t)]$$

I know that the expected value of the Brownian motion is $0$, so this is a covariance, but the absolute value make me insecure on how calculate the result.

Answer 1

$B$ has the same distribution as $-B$, so $$J:=E[|B(t)|* B(1-t)]= E[|-B(t)|* (-B(1-t))]=-J.$$ Therefore $J=0$.