---
title: "Gauss 2-point and 3-point: log(x+1)+sin 2x"
subject: "Numerical Method"
chapter: 5
key: "semester_4/numerical_method:gauss-2pt-3pt-log-sin"
source: https://www.bctengineer.com/q/semester_4/numerical_method/gauss-2pt-3pt-log-sin
site: BCT Engineering
---

# Gauss 2-point and 3-point: log(x+1)+sin 2x

Numerical Method · Chapter 5: Numerical Differentiation and Integration

- Asked in: 2068 Bhadra, 6 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/numerical_method/gauss-2pt-3pt-log-sin
- Whole chapter: https://www.bctengineer.com/md/semester_4/numerical_method/chapter-5

## Question

Evaluate the integral I = ∫ from 0.2 to 1.4 of (log(x + 1) + sin 2x) dx using Gaussian 2 point and 3 point formula. [limits partly illegible in the scan]

## Short answer

With Gauss-Legendre quadrature after mapping x=0.6t+0.8: 2-point gives ≈1.6056, 3-point gives ≈1.6141.

## Answer

Assumptions: log means the natural logarithm ln, sin 2x uses radians, and the limits are 0.2 to 1.4 (partly illegible in the scan).

#### Change of limits
x = [(b−a)t + (b+a)]/2 = [(1.4−0.2)t + (1.4+0.2)]/2 = 0.6t + 0.8, dx = 0.6 dt.

I = 0.6 ∫₋₁¹ F(t) dt, where F(t) = ln(1.8 + 0.6t) + sin(1.6 + 1.2t).

#### 2-point Gauss–Legendre (t = ±0.577350, w = 1, 1)
| t | weight | x = 0.6t+0.8 | f(x) | w·f(x) |
|---|---|---|---|---|
| -0.5774 | 1.0000 | 0.4536 | 1.1618 | 1.1618 |
| 0.5774 | 1.0000 | 1.1464 | 1.5143 | 1.5143 |

```text
I₂ = 0.6 × (2.6761) = 1.6056
```

#### 3-point Gauss–Legendre (t = 0, ±0.774597; w = 8/9, 5/9, 5/9)
| t | weight | x = 0.6t+0.8 | f(x) | w·f(x) |
|---|---|---|---|---|
| -0.7746 | 0.5556 | 0.3352 | 0.9105 | 0.5058 |
| 0.0000 | 0.8889 | 0.8000 | 1.5874 | 1.4110 |
| 0.7746 | 0.5556 | 1.2648 | 1.3920 | 0.7734 |

```text
I₃ = 0.6 × (2.6902) = 1.6141
```

**Answer: 2-point result ≈ 1.6056; 3-point result ≈ 1.6141** (exact value 1.6140; the 3-point result is closer).

## Check yourself

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

1. How is log interpreted here?
   A. Base-10 log
   B. Base-2 log
   C. log of e
   D. Natural logarithm ln
   Answer: D. The answer assumes log = ln. (Key point: Log meaning)

2. What is the change of variable on [0.2, 1.4]?
   A. x = 0.6t + 0.8, dx = 0.6 dt
   B. x = 1.2t + 1.6
   C. x = 0.8t + 0.6
   D. x = t + 0.8
   Answer: A. (b-a)/2 = 0.6, (b+a)/2 = 0.8. (Key point: Substitution)

3. Which nodes and weights are used for the 3-point rule?
   A. t = ±0.57735; w = 1
   B. t = 0, ±0.774597; w = 8/9, 5/9, 5/9
   C. t = 0, ±1; w = 4/3
   D. t = ±0.774597; w = 1
   Answer: B. Standard 3-point Gauss-Legendre. (Key point: 3-point nodes)

4. Which is closer to exact 1.6140?
   A. 2-point, 1.6056
   B. Both equal
   C. 3-point, 1.6141
   D. Neither
   Answer: C. More points give higher exactness degree. (Key point: Comparison)

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