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After applying $n$ times the Poincaré Lemma, we get

$$ H^k(\mathbb{R} ^n)\simeq \left\{ \begin{array}{rl} \mathbb{R} & k=0\\ 0 & k>0 \end{array}\right. \qquad \qquad \qquad H^k_c(\mathbb{R} ^n)\simeq \left\{ \begin{array}{rl} 0 & k < n\\ \mathbb{R} & k=n \end{array}\right. $$