Poincaré lemma and $k$-forms

Question

(bad English, will keep myself short)
I am trying to show that with the Poincaré lemma ${ d \alpha= \omega }$ is true for
$${ \alpha := \sum_{i_1 \lt ... \lt i_k} \sum_{j=1}^{k}{(-1)^{j+1}\left( \int_{0}^{1}{t^{k-1} \omega_{i_1 \cdots i_k}(tx)\,dt}\right)x_{i_j}\,dx_{i_1} \land ... \land{\widehat{dx_{i_j}} \land ... \land dx_{i_k} } }}.$$
The roof means the term is omitted.
(I don't know if it is necessary but this is the Poincaré lemma definition that I have: Let ${U \subset \mathbb{R}^n }$ open set and star-shaped regarding ${ 0 \in \mathbb{R}^n }$ , then
${\omega= \sum_{i_1 \lt ... \lt i_k}\omega_{i_1...i_k}dx_{i_1} \land ... \land dx_{i_k} }$
is a ${ C^{ \infty } }$ $k$-Form on $U$ with ${ d \omega=0 }$)