---
title: "Half range sine series: x"
subject: "Mathematics III"
chapter: 4
key: "semester_3/mathematics_iii:half-range-sine-series-x-three"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/half-range-sine-series-x-three
site: BCT Engineering
---

# Half range sine series: x

Mathematics III · Chapter 4: Fourier Series

- Asked in: 2081 Chaitra (new course), 2 marks
- Video: [Half range sine series/ Fourier series/ PU 2018/ Engineering mathematics 3/ Third semester](https://www.youtube.com/watch?v=s__Bkco7Upg) — Rupak Deo
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/half-range-sine-series-x-three
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-4

## Question

Find the half range Fourier sine series of the function f(x) = x in the interval (0, 2).

## Short answer

The half range sine series of f(x)=x on (0,2) has coefficients b_n = (4/(nπ))(-1)^(n+1). Therefore, x = 4/π Σ_{n=1}^∞ [(-1)^(n+1)/n] sin(nπx/2) for 0<x<2.

## Answer

The half range sine series on (0, L) is f(x) = Σ bₙ sin(nπx/L), with bₙ = (2/L)∫₀ᴸ f(x) sin(nπx/L) dx. Here L = 2.

```text
bₙ = ∫₀² x sin(nπx/2) dx
   = [ −(2x/nπ) cos(nπx/2) ]₀² + (2/nπ)∫₀² cos(nπx/2) dx
   = −(4/nπ) cos nπ + 0
   = 4(−1)ⁿ⁺¹/(nπ)
```

**Answer:**

x = (4/π) Σ (n=1 to ∞) [(−1)ⁿ⁺¹/n] sin(nπx/2),  0 < x < 2

= (4/π)[ sin(πx/2) − ½ sin(πx) + ⅓ sin(3πx/2) − … ]

## Check yourself

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

1. With L = 2, bn =
   A. (1/2)∫
   B. ∫0^2 x sin(nπx/2) dx
   C. ∫ x cos
   D. (2/π)∫
   Answer: B. 2/L = 1. (Key point: Formula)

2. bn =
   A. 4(−1)^n/(nπ)
   B. 2/(nπ)
   C. 4(−1)^(n+1)/(nπ)
   D. 0
   Answer: C. −4cos nπ/(nπ). (Key point: bn)

3. x =
   A. (4/π)[sin + ½ sin + ...]
   B. (2/π)[...]
   C. (4/π)[cos ...]
   D. (4/π)[sin(πx/2) − ½ sin πx + ⅓ sin(3πx/2) − ...]
   Answer: D. Alternating. (Key point: Series)

## Close variants set in other exams

- [Half range sine series: e^(ax)](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahalf-range-sine-series-e-ax) — Mathematics III, chapter 4, asked 5×
- [Half range sine series: x²](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahalf-range-sine-series-x2-two) — Mathematics III, chapter 4, asked 9×
- [Half range sine series: x − x²](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahalf-range-sine-series-x-x2-three) — Mathematics III, chapter 4, asked 11×
- [Half range sine series: eˣ](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahalf-range-sine-series-e-two) — Mathematics III, chapter 4, asked 6×

## Related in Mathematics III

- [Half range sine series: x²](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahalf-range-sine-series-x2-three) — Mathematics III, chapter 4, asked 9×
- [Half range sine series: π − x](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahalf-range-sine-series-x) — Mathematics III, chapter 4, asked 13×
- [Half range sine series: πx − x²](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahalf-range-sine-series-x-x2-two) — Mathematics III, chapter 4
- [Half range sine series: x − x²](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahalf-range-sine-series-x-x2) — Mathematics III, chapter 4, asked 14×
- [Half range sine series: eˣ](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahalf-range-sine-series-e) — Mathematics III, chapter 4, asked 12×
- [Half range sine series: πx − x²](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahalf-range-sine-series-x-x2-four) — Mathematics III, chapter 4, asked 12×

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