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I am struggling to prove this, $$ \int \mathcal{N}_\mathbf{x}(\mu,\Sigma)e^{a^T\mathbf{x}}d\mathbf{x} = e^{{a^T\mu}+\frac 12a^T\Sigma a} $$

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Hi, and welcome to Math.StackExchange. Your question is lacking in context. In an edit to this question you should provide the source in which this integral is stated. Also, for us to not rework anything you have done already, I suggest you edit the question with your progress so far. – Karl Kronenfeld May 20 '13 at 8:23

Complete the square in the exponent, that is, integrate the identity $$ \mathcal{N}_\mathbf x(\mu,\Sigma)\cdot\mathrm e^{a^T\mathbf x}=\mathrm e^{a^T\mu+\frac 12a^T\Sigma a}\cdot\mathcal{N}_\mathbf x(\mu+\Sigma a,\Sigma). $$

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