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 \( [0,\infty) \).