Discussion
The exact solution of \(\int_{0}^{4} \frac{dx}{1+x}\) is represented as \(n\).
If \(m\) represents numerically evaluated value of the above integral using Trapezoidal rule by considering four equal subintervals in the range of \(x\), then \((m - n)\) is
Answer: A
The difference \(m - n\) is 0.074.
Exactly, \(n = \ln 5 = 1.6094\). With four intervals, \(h = 1\) and the ordinates at \(x = 0, 1, 2, 3, 4\) are \(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}\):
\[ \begin{aligned} m &= \tfrac{1}{2}\left(1 + \tfrac{1}{5}\right) + \tfrac{1}{2} + \tfrac{1}{3} + \tfrac{1}{4} \\ &= 0.6 + 1.0833 = 1.6833 \end{aligned} \]
So \(m - n = 1.6833 - 1.6094 = 0.074\). The sign is positive because \(1/(1+x)\) is convex, and chords lie above a convex curve, so the trapezoidal rule overestimates; -0.074 has the wrong sign, and 0.003 or -0.003 is far smaller than the trapezoidal error with only four intervals.
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