$$ -\int_{}^{} \frac {exp(\frac{1}{2}x^TP^{-1}x)}{{(2\pi)}^{\frac{1}{2}}|P|^{\frac{1}{2}}}log_2(\frac {exp(\frac{1}{2}x^TP^{-1}x)}{{(2\pi)}^{\frac{1}{2}}|P|^{\frac{1}{2}}}) dx$$ where x is a N dimensional Gaussian vector having variance as P and mean zero. My approach is let be given by $$x=\begin{bmatrix} x_{1} \\ x_{2} \\ \vdots\\ x_{n} \end{bmatrix}$$ and P by \begin{bmatrix} \sigma_{1} & 0 & 0 & \dots & 0 \\ 0 & \sigma_{2} & 0 & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & \sigma_{n} \end{bmatrix} I am getting an integral like$$-\int_{}^{} \frac {exp(\frac{1}{2}(\frac{x_1^2}{\sigma_1^2}+\frac{x_2^2}{\sigma_2^2}+\frac{x_3^2}{\sigma_3^2}---+\frac{x_n^2}{\sigma_n^2}))}{{(2\pi)}^{\frac{1}{2}}|P|^{\frac{1}{2}}}log_2(\frac {exp(\frac{1}{2}(\frac{x_1^2}{\sigma_1^2}+\frac{x_2^2}{\sigma_2^2}+\frac{x_3^2}{\sigma_3^2}---+\frac{x_n^2}{\sigma_n^2}))}{{(2\pi)}^{\frac{1}{2}}|P|^{\frac{1}{2}}}) dx$$ but confused how to do this dx stuff when x is a vector. Am I doing it right?
The answer given is $$\frac{n}{2}log_2(2 \pi e)+ \frac {1}{2} log_2|P|$$