This gives us the new random variable $y$ in the domain $y \in [0,\infty)$ determined through the random variable $x \in [0,1]$ generated by functions like $ran0$.

This means that if we can factor out $\exp{(-y)}$ from an integrand we may have $$\begin{equation*} I=\int_0^{\infty}F(y)dy=\int_0^{\infty}\exp{(-y)}G(y)dy \end{equation*}$$ which we rewrite as $$\begin{equation*} \int_0^{\infty}\exp{(-y)}G(y)dy= \int_0^{1}G(y(x))dx\approx \frac{1}{N}\sum_{i=1}^NG(y(x_i)), \end{equation*}$$ where $x_i$ is a random number in the interval $[0,1]$.