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

Step 1: Find the exact value \( n \).
\[ n = \int_0^4 \frac{dx}{1+x} = \Big[\ln(1+x)\Big]_0^4 = \ln 5 - \ln 1 = \ln 5 \]
Numerically, \( \ln 5 = 1.60944 \) to five decimal places.

Step 2: Set up the Trapezoidal rule with four subintervals.
The range \( x = 0 \) to \( 4 \) split into four equal parts gives step size \( h = 1 \), with nodes \( x = 0, 1, 2, 3, 4 \).
The function values are \( f(0) = 1 \), \( f(1) = 0.5 \), \( f(2) = 0.33333 \), \( f(3) = 0.25 \), \( f(4) = 0.2 \).

Step 3: Apply the Trapezoidal formula.
\[ m = \frac{h}{2}\Big[f(0) + 2\big(f(1)+f(2)+f(3)\big) + f(4)\Big] = \frac{1}{2}\Big[1 + 2(1.08333) + 0.2\Big] = \frac{1}{2}(3.36667) = 1.68333 \]

Step 4: Compute \( m - n \).
\[ m - n = 1.68333 - 1.60944 = 0.07389 \approx 0.074 \]

Final Answer:
The Trapezoidal estimate slightly overshoots the true log value, giving a small positive error. \[ \boxed{m - n \approx 0.074} \]