I want to show that for each $1 \le k\le n$ we have $$ H_{dR}^k(\mathbb{R}^n)=0 $$
The strategy is to construct for each $k$ a linear map $$h_k:\Omega^k(\mathbb{R}^n)\to \Omega^{k-1}(\mathbb{R}^n)$$ such that $$ d_{k-1}\circ h_k + h_{k+1}\circ d_{k}=Id_{\Omega^k(\mathbb{R}^n)} \qquad\qquad[1]$$ where $d_k:\Omega^{k}(\mathbb{R}^n)\to \Omega^{k+1}(\mathbb{R}^n)$ is the exterior derivative.
I have the following definition: for each $\omega=\omega_{i_1\dots i_k}dx^{i_1}\wedge\dots\wedge dx^{i_k}$ in $\Omega^k(\mathbb{R}^n)$ we define $$h_k(\omega)(x)=k \left[ \int_0^1t^{k-1} \omega_{i_1\dots i_k}(tx) \;dt\right]x^{i1} dx^{i_2}\wedge\dots\wedge dx^{i_k}$$
Can you help me to show that this definition satisfies $[1]$?
For example I have $$d_k(w)=\sum_j \frac{\partial(\omega_{i_1\dots i_k})}{\partial x^j} dx^j\wedge dx^{i_1}\wedge\dots\wedge dx^{i_k}$$ and
$$ d_{k-1}( h_k (w))(x)=k\sum_j \frac{\partial(\left[ \int_0^1t^{k-1} \omega_{i_1\dots i_k}(tx) \;dt\right]x^{i1})}{\partial x^j} dx^j\wedge dx^{i_2}\wedge\dots\wedge dx^{i_k}$$ and
$$ h_{k+1}(d_k(\omega))(x)=(k+1)\left[ \int_0^1t^{k} \frac{\partial(\omega_{i_1\dots i_k})}{\partial x^j} (tx) \;dt\right]x^{j}dx^{i_1}\wedge\dots\wedge dx^{i_k}$$
I don't know if my calculations above are correct, and even if they are, I can't go on.
Please give most details you can if you do sofisticated manipulations/calculatons.