The authors find the explicit solutions of the best \( L^1(\mathbb{R}) \) (unrestricted and one-sided) approximation problems by entire functions of exponential type of at most \( \pi \) of the truncated and the odd Gaussians \[ G_{\lambda}^+(x)=x_+^0e^{-\pi\lambda x^2},\quad G_{\lambda}^o(x)=\mathrm{sign}(x)e^{-\pi\lambda x^2}. \] For example, the best \( L^1(\mathbb{R}) \) approximation of \(G_{\lambda}^+ \) is \[ K_{\lambda}^+(z)=\frac{\sin\pi z}{\pi}\sum_{n=1}^{\infty}(-1)^n\left\{\frac{G_{\lambda}^+(n)}{z-n}-\frac{G_{\lambda}^+(n)}{n}\right\}. \] The second part of the paper is devoted to the integration on the free parameter \( \lambda \) as a tool to generate the solution of these best approximation problems for a class of truncated and odd functions.