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:

\(x\) | \(3.0\) | \(3.5\) | \(4.0\) | \(4.5\) | \(5.0\) |

\(x \ln x\) | \(3.296\) | \(4.385\) | \(5.545\) | \(6.768\) | \(8.047\) |