In this question, I've already done the item a) by following the hint. I'm struggling (even with the hint) to calculate item b). I believe I have to follow the steps in the proof of Poincaré's Lemma, however I couldn't obtain the integration of $H^*w$ as indicated in the hint. Would someone help me in how to obtain the expression in item b)? Thanks in advance.
I've done some progress, but I coudn't find the final result.
MY ATTEMPT: Considering the contraction
$H: \mathbb{R}^3\times \mathbb{R}\rightarrow \mathbb{R}^3$ given by $H(p,t)=(tx, ty, tz)$.
Calculating the differential, we have
$dH=(xdt+tdx, ydt+tdy, zdt+tdz)$.
Now remembering that $\omega=Ady\wedge dz + Bdz\wedge dx+ Cdx\wedge dy$ in item a), the pullback is
$\begin{align*} \overline{\omega}&= H^*\omega=\omega (dH)=\\ &=A(tx, ty, tz)(ydt+tdy)\wedge(zdt+tdz)+\\ &+B(tx, ty, tz)(zdt+tdz)\wedge(xdt+tdx)+\\ &+C(tx, ty, tz)(xdt+tdx)\wedge(ydt+tdy)\\ \end{align*}$
$\begin{align*} &=A(tx, ty, tz)(tydt\wedge dz+tzdy\wedge dt+t^2dy\wedge dz)\\ &+B(tx, ty, tz)(zt dt\wedge dx+txdz\wedge dt + t^2 dz^dx)\\ &+C(tx, ty, tz)(xtdt\wedge dy+ tydx\wedge dt+t^2dx\wedge dy) \end{align*}$