---
title: "Gauss 3-point: cos2x/(1+sin x)"
subject: "Numerical Method"
chapter: 5
key: "semester_4/numerical_method:gauss-3pt-cos2x-over-1-sinx"
source: https://www.bctengineer.com/q/semester_4/numerical_method/gauss-3pt-cos2x-over-1-sinx
site: BCT Engineering
---

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

Numerical Method · Chapter 5: Numerical Differentiation and Integration

- Asked in: 2070 Magh, 5 marks
- Video: [GAUSSIAN INTEGRATION // Gauss Legendre 1, 2, 3 Point Formula // Gauss Quadrature 2,3,4 point Formula](https://www.youtube.com/watch?v=YzSdt1_kIJg) — Arya Anjum
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/numerical_method/gauss-3pt-cos2x-over-1-sinx
- Whole chapter: https://www.bctengineer.com/md/semester_4/numerical_method/chapter-5

## Question

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

## 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.

## Answer

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).

#### Change of limits
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).

#### Nodes and function values
| 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 |

```text
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).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What substitution maps [2, 3] to [-1, 1]?
   A. x = t + 2.5, dx = dt
   B. x = 0.5t + 2.5, dx = 0.5 dt
   C. x = 0.5t + 2, dx = 0.5 dt
   D. x = 2.5t + 0.5, dx = 2.5 dt
   Answer: B. (b-a)/2 = 0.5 and (b+a)/2 = 2.5. (Key point: Substitution)

2. In which unit are the angles evaluated?
   A. Degrees
   B. Grads
   C. Radians
   D. Either; no difference
   Answer: C. Calculus integrals use radians. (Key point: Angle unit)

3. Which weights multiply f at t = -0.7746, 0, 0.7746?
   A. 1, 1, 1
   B. 8/9, 5/9, 8/9
   C. 1/3, 1/3, 1/3
   D. 5/9, 8/9, 5/9
   Answer: D. The middle node has the bigger weight. (Key point: Weights)

4. What is the result?
   A. 0.5 × 0.4054 = 0.2027
   B. 0.4054
   C. 0.2027 × 0.5 = 0.1014
   D. 0.5 + 0.4054 = 0.9054
   Answer: A. The weighted sum is multiplied by 0.5. (Key point: Result)

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