We denote $\varphi$ the function defined by: $$\begin{cases} \varphi(x) = 0 & \text{if } |x| \geqslant 1 \\ \varphi(x) = \exp\left(-\frac{x^2}{1-x^2}\right) & \text{if } |x| < 1 \end{cases}$$ a) Show that $\int_{\mathbb{R}} \varphi(t) \mathrm{d}t$ is a strictly positive real number. b) For every real number $x$, we set $\theta(x) = \frac{\varphi(x)}{\int_{\mathbb{R}} \varphi(t) \mathrm{d}t}$ and, for every non-zero natural number $n$, $\rho_n(x) = n\theta(nx)$. Show that $$\forall n \in \mathbb{N}^* \quad \int_{\mathbb{R}} \rho_n(x) \mathrm{d}x = 1$$
We denote $\varphi$ the function defined by:
$$\begin{cases} \varphi(x) = 0 & \text{if } |x| \geqslant 1 \\ \varphi(x) = \exp\left(-\frac{x^2}{1-x^2}\right) & \text{if } |x| < 1 \end{cases}$$
a) Show that $\int_{\mathbb{R}} \varphi(t) \mathrm{d}t$ is a strictly positive real number.\\
b) For every real number $x$, we set $\theta(x) = \frac{\varphi(x)}{\int_{\mathbb{R}} \varphi(t) \mathrm{d}t}$ and, for every non-zero natural number $n$, $\rho_n(x) = n\theta(nx)$.
Show that
$$\forall n \in \mathbb{N}^* \quad \int_{\mathbb{R}} \rho_n(x) \mathrm{d}x = 1$$