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<math>
<math>
\begin{aligned}
\begin{aligned}
\int_{-\frac{L}{2}}^{\frac{L}{2}}\int_{f([-\frac{L}{2},\frac{L}{2}])}p(x)p(y|x)\ln p(y|x)dydx\\
\int_{-\frac{L}{2}}^{\frac{L}{2}}\int_{f([-\frac{L}{2},\frac{L}{2}])}p(x)p(y|x)\ln p(y|x)dydx\approx \int_{-\infty}^{\infty}\int_{\infty}^{\infty}p(x)p(y|x)\ln p(y|x)dydx\\
&=\int_{-\infty}^{\infty}\int_{\infty}{\infty}\frac{1}{L}\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{(y-f(x))^2}{\sigma^2}\right)\ln\left[\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{(y-f(x))^2}{\sigma^2}\right)\right]dydx\\
&=\int_{-\infty}^{\infty}\int_{\infty,\infty])}\frac{1}{L}\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{(y-f(x))^2}{\sigma^2}\right)\ln\left[\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{(y-f(x))^2}{\sigma^2}\right)\right]dydx\\
&=\ln(\frac{L}{\sqrt{2\pi e}})
&=\ln(\frac{L}{\sqrt{2\pi e}})
\end{aligned}
\end{aligned}