- Asked 3/13
- 2075 Baisakh · 6 marks
Gauss 3-point: e^(sin x)
Short answer
Transform to [-1,1] with x = (π/4)t + π/4, dx = π/4 dt. Apply 3-point Gauss-Legendre with nodes ±√0.6,0 and weights 5/9,8/9,5/9 to get I ≈ 3.1039.
Evaluate ∫ from 0 to π/2 of e^(sin x) dx using Gaussian 3-point formula.
The 3-point Gauss–Legendre formula works on [−1, 1]:
∫₋₁¹ g(t) dt ≈ (5/9)·g(−√0.6) + (8/9)·g(0) + (5/9)·g(√0.6), √0.6 = 0.774597
Change of limits
For ∫ₐᵇ f(x) dx put x = (b−a)/2·t + (b+a)/2. Here a = 0, b = π/2 = 1.570796, so
x = 0.785398·t + 0.785398, dx = 0.785398 dt
∫ f(x) dx = 0.785398 ∫₋₁¹ f(0.785398t + 0.785398) dt
Nodes and function values
| t | x = 0.7854t + 0.7854 | f(x) | weight w | w·f(x) |
|---|---|---|---|---|
| -0.774597 | 0.177031 | 1.192567 | 5/9 | 0.662537 |
| 0.000000 | 0.785398 | 2.028115 | 8/9 | 1.802769 |
| 0.774597 | 1.393765 | 2.676128 | 5/9 | 1.486738 |
I ≈ 0.785398 × [ 3.952044 ] = 3.1039
(The accurate value is 3.1044, so the 3-point result is good to about three significant figures.)
Answer: I ≈ 3.1039
✓ Answer checked by Claude Opus 5.5 (high effort)
Related questions
- Gauss 3-point: sin x³ + cos x² · Numerical Method, chapter 5 · asked 2×
- Gauss 3-point: 1/(x²+2x+1) · Numerical Method, chapter 5 · asked 3×
- Gauss 2-point and 3-point: log(x+1)+sin 2x · Numerical Method, chapter 5
- Gauss 3-point: cos2x/(1+sin x) · Numerical Method, chapter 5
- Gauss 3-point: e^(-x²) · Numerical Method, chapter 5 · asked 2×
- Gauss-Legendre 2-point: e^(-x²) · Numerical Method, chapter 5
- Laplace: five-point formula, 2x2 · Numerical Method, chapter 7
- Fixed point: sin x + 3x - 2 · Numerical Method, chapter 2 · asked 2×
- DUIS: ∫ e^(−x) sin bx / x dx · Mathematics I, chapter 2 · asked 10×
- DUIS: ∫ log((a+b sin x)/(a−b sin x)) dx/sin x · Mathematics I, chapter 2 · asked 9×
- DUIS: ∫ e^(−ax) sin x / x dx · Mathematics I, chapter 2 · asked 8×
- Evaluate using Gauss' divergence theorem · Mathematics III, chapter 2 · asked 11×