added the part of integration.

integration.tex

 \subsection{Newton–Cotes formulas}
+\label{sec:int_newton_cotes}
+
 We derived the following formulas for evaluating the following integral numerically: $I = \int_{x_1}^{x_2} f(x) \, dx.$
 \begin{eqnarray}
 \mathit{Midpoint~rule:} & I = h \, f(\frac{x_1+x_2}{2})  + \mathcal{O}\,(h^3) \\
@@ -15,10 +17,9 @@ where $h := x_2 - x_1 = x_1 - x_0$ is the width of one slice. Note that the erro
 \begin{equation}
 I = \int_{-1}^{1} e^{-x^2} dx.
 \end{equation}
-You may use the following SciPy functions: \href{https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.trapezoid.html#scipy.integrate.trapezoid}{\texttt{scipy.integrate.trapezoid}} and \href{https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.simpson.html#scipy.integrate.simpson}{\texttt{scipy.integrate.simpson}}. % and \href{https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.romb.html}{\texttt{scipy.integrate.romb}}.
+You may use the following SciPy functions: \href{https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.trapezoid.html#scipy.integrate.trapezoid}{\texttt{scipy.integrate.trapezoid}} and \href{https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.simpson.html#scipy.integrate.simpson}{\texttt{scipy.integrate.simpson}}.
 \item Repeat the calculation using three different values of $h$. Plot the error $\Delta I$ against $h$. Show that the numerical estimates converge to the exact value according to the expected order of convergence. First, you can plot the error [which will be $\mathcal(O)(h^n)]$ against $h$ as a log-log plot and look at the slope. This will give you an idea of the order of convergence $n$. Then, you can actually compute $n$ using Eq.\eqref{eq:order_convg}.
 
-
 \item Using the Trapezoidal and Simpson's methods, compute the energy loss due to GW emission from binary black holes using the numerical-relativity data discussed in Sec.~\ref{sec:BBH_nr_data_fd}. The radiated energy can be computed from the GW polarizations as 
 \begin{equation}
 E = \int_{-\infty}^{\infty} \left[\left(\frac{dh_+}{dt}\right)^2 +  \left(\frac{dh_\times}{dt}\right)^2 \right] \, dt. 
@@ -27,4 +28,14 @@ E = \int_{-\infty}^{\infty} \left[\left(\frac{dh_+}{dt}\right)^2 +  \left(\frac{
 \end{enumerate}
 
 
+\subsection{Romberg method, Adaptive quadrature, Gaussian quadrature}
+
+\subsubsection*{Problems:}
+\begin{enumerate}
+\item Repeat problem 3 of Sec.\ref{sec:int_newton_cotes} using the adaptive Gaussian quadrature. You may use the function \href{https://docs.scipy.org/doc//scipy-1.13.0/reference/generated/scipy.integrate.quad.html#scipy.integrate.quad}{\texttt{scipy.integrate.quad}}.
+\item Repeat problem 5 of Sec.\ref{sec:int_newton_cotes} using the Romberg's method. You may use the function \href{https://docs.scipy.org/doc//scipy-1.13.0/reference/generated/scipy.integrate.romb.html#scipy.integrate.romb}{\texttt{scipy.integrate.romb}}.
+
+	
+
+\end{enumerate}