I am working out some of the properties for the Ito integral with Brownian motion and I am trying to use the definition to verify that $$ \int _0 ^t s \, dB_s = tB_t - \int _0 ^t B_s\, ds $$ and
$$ \int _0 ^t B_s^2 \, dB_s = \frac{1}{3}(B_t^3) - \int _0 ^t B_s\, ds $$
I am having trouble thinking about how to attack this so any help or suggestions on how to get started would be great!
EDITED to meet edit of question
The first equation is (after the edit) true. Consider the twodimensional continuous semimartingale $\left( t,B_t\right)$, and function $f(x,y)=xy$ we get $$D_xf(x,y)=y\quad D_yf(x,y)=x\quad D_1D_1f=D_2D_2f=0\quad D_1D_2f=D_2D_1f=1$$ And therefore ITO's formula gives $$tB_t=0+\int_0^t s\; dB_s+\int_0^t B_s\; ds.$$
The 2nd equation is (after the edit) true as well. You can convince yourself with an argument similar to the one i gave. Apply the 1-dimensional ITO's formula on $B_t$ with $f(x)=x^3$