Gauss 3-point: cos2x/(1+sin x)

Short answer

Transform the interval to [-1,1] using x = 0.5t + 2.5 and apply the three-point Gauss-Legendre formula. The integral is approximately 0.2027.

Evaluate ∫ from 2 to 3 of cos 2x/(1 + sin x) dx using Gauss quadrature three-point formula.

Gauss–Legendre 3-point formula on [−1, 1]: ∫₋₁¹ F(t) dt ≈ (5/9)F(−0.774597) + (8/9)F(0) + (5/9)F(0.774597).

For [a, b] = [2, 3] put x = [(b−a)t + (b+a)]/2 = 0.5t + 2.5, so dx = 0.5 dt:

I = 0.5 ∫₋₁¹ f(0.5t + 2.5) dt, with f(x) = cos 2x / (1 + sin x) (x in radians).

t	weight	x = 0.5t+2.5	f(x)	w·f(x)
-0.7746	0.5556	2.1127	-0.2520	-0.1400
0.0000	0.8889	2.5000	0.1775	0.1577
0.7746	0.5556	2.8873	0.6979	0.3877

I ≈ 0.5 × [ -0.1400 + 0.1577 + 0.3877 ]
  = 0.5 × 0.4054
  = 0.2027

Answer: ∫₂³ cos2x/(1+sin x) dx ≈ 0.2027 (the exact value is 0.2027).

✓ Answer checked by Claude Opus 5.5 (high effort)

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