Question

The exact solution of \( \displaystyle\int_0^4 \frac{dx}{1+x} \) is represented as \( n \).
If \( m \) represents the numerically evaluated value of the above integral using the Trapezoidal rule by considering four equal subintervals in the range of \( x \), then \( (m-n) \) is

• A. \( 0.074 \)
• B. \( -0.074 \)
• C. \( -0.003 \)
• D. \( 0.003 \)

Hint:

Compute the exact log value first, then apply the Trapezoidal formula with five nodes spaced 1 apart.

Answer 1

The Correct Option is A

This question compares an exact integral against its Trapezoidal rule estimate, so two separate calculations are needed before subtracting them.

For the exact part, $\int_0^4 dx/(1+x)$ is a standard log integral, giving $n = \ln(1+x)$ evaluated from 0 to 4, which is $\ln 5$, about 1.6094.

For the estimate, the Trapezoidal rule with four equal strips over $[0,4]$ uses step size $h = 1$ and the five function values at $x = 0, 1, 2, 3, 4$: these are $1$, $0.5$, $1/3$, $0.25$, and $0.2$. The rule weights the two end values once and the three interior values twice, then multiplies by $h/2$:

$m = 0.5 \times \left[1 + 0.2 + 2(0.5 + 0.33333 + 0.25)\right] = 0.5 \times \left[1.2 + 2.16667\right] = 0.5 \times 3.36667 = 1.68333$.

The curve $1/(1+x)$ bends downward, so the straight trapezoid edges sit slightly above the curve between nodes, and the Trapezoidal estimate comes out a little higher than the true value, a small positive error rather than negative.

Subtracting, $m - n = 1.68333 - 1.60944 = 0.0739$, which rounds to $0.074$, matching option (A) and ruling out the negative sign options.