Mapping integration points and weights

If we have an integral on the form $$ \begin{equation*} \int_0^{\infty}f(t)dt, \end{equation*} $$ we can choose new mesh points and weights by using the mapping $$ \begin{equation*} \tilde{x}_i=tan\left\{\frac{\pi}{4}(1+x_i)\right\}, \end{equation*} $$ and $$ \begin{equation*} \tilde{\omega}_i= \frac{\pi}{4}\frac{\omega_i}{cos^2\left(\frac{\pi}{4}(1+x_i)\right)}, \end{equation*} $$ where \( x_i \) and \( \omega_i \) are the original mesh points and weights in the interval \( [-1,1] \), while \( \tilde{x}_i \) and \( \tilde{\omega}_i \) are the new mesh points and weights for the interval \( [0,\infty) \).