Use the Composite Trapezoidal, Simpson's and Midpoint Rules to approximate the integral ∫12 x ln⁡(x)dx, n=4 (For the Midpoint Rule use n + 2 sub intervals.)

a) Trapezoidal Rule: The interval [1, 2 can be partitioned into 4 sub-intervals, P={1, 54, 32, 74, 2}, where: Δ x=b − a4=14. ∫12 xln⁡(x)dx=14{f(1) + 2f(54) + 2f(32) + 2f(74) + f(2)} 14{(ln⁡(1)) + 2(54 ln⁡(54)) + 2(32ln⁡(32)) + 2(74ln⁡(74)) + 2(2ln⁡(2))} 14{0 + 0.55788589 + 1.2163953 + 1.9586553 + 2.7725887}=1.6263813 b) Simpsons

Answered: "Use the Composite Trapezoidal, Simpson's and Midpoint Rules..."

aflacatn

Answered question

2020-12-12

Use the Composite Trapezoidal, Simpson's and Midpoint Rules to approximate the integral
$\int_{1}^{2} x \ln (x) d x, n = 4$
(For the Midpoint Rule use $n + 2$ sub intervals.)

Answer & Explanation

gwibdaithq

Skilled2020-12-13Added 84 answers

a) Trapezoidal Rule:
The interval [1, 2 can be partitioned into 4 sub-intervals,

P = {1, \frac{5}{4}, \frac{3}{2}, \frac{7}{4}, 2},

where:

Δ x = \frac{b - a}{4} = \frac{1}{4} .

\int_{1}^{2} x \ln (x) d x = \frac{1}{4} {f (1) + 2 f (\frac{5}{4}) + 2 f (\frac{3}{2}) + 2 f (\frac{7}{4}) + f (2)}

\frac{1}{4} {(\ln (1)) + 2 (\frac{5}{4} \ln (\frac{5}{4})) + 2 (\frac{3}{2} \ln (\frac{3}{2})) + 2 (\frac{7}{4} \ln (\frac{7}{4})) + 2 (2 \ln (2))}

\frac{1}{4} {0 + 0.55788589 + 1.2163953 + 1.9586553 + 2.7725887} = 1.6263813

b) Simpsons

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