The continuous function \(f\) maps the real interval \[\begin{equation*} a \le x \le b \end{equation*}\]

into a subset of the positive real numbers. Derive the trapezium rule for the approximate evaluation of the area under the curve \(y = f(x)\) between \(x=a\) and \(x=b\), using \((n-1)\) points of subdivision at a distance \(h = (b-a)/n\) apart.

Hence obtain an approximation to the integral \[\begin{equation*} I = \int_3^5 x \ln x \:dx \end{equation*}\]

using the points of subdivision given in the following table:

\(x\) \(3.0\) \(3.5\) \(4.0\) \(4.5\) \(5.0\)
\(x \ln x\) \(3.296\) \(4.385\) \(5.545\) \(6.768\) \(8.047\)