---
title: "Half range sine series: lx − x²"
subject: "Mathematics III"
chapter: 4
key: "semester_3/mathematics_iii:half-range-sine-series-lx-x2"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/half-range-sine-series-lx-x2
site: BCT Engineering
---

# Half range sine series: lx − x²

Mathematics III · Chapter 4: Fourier Series

- Asked in: 2076 Asoj, 5 marks
- Video: [VTU Half Range Fourier Series | f(x) = lx − x² | Sine Series | EE301, EC301, ME301](https://www.youtube.com/watch?v=OrjVOptEICE) — Mathematics Tutor
- 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-lx-x2
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-4

## Question

If f(x) = lx − x² in (0, l), show that the half range sine series for f(x) is (8l²/π³) Σ_{n=0}^∞ [1/(2n+1)³] sin((2n+1)πx/l).

## Short answer

b_n = (2/l)∫_0^l (lx - x²) sin(nπx/l) dx = 4l²(1-(-1)^n)/(n³π³). For even n it vanishes; for n=2m+1, b_n = 8l²/(π³(2m+1)³). The half-range sine series is (8l²/π³) ∑_{m=0}^∞ sin((2m+1)πx/l)/(2m+1)³.

## Answer

**To show:** for f(x) = lx − x² on (0, l), the half-range sine series is (8l²/π³) Σ_{n=0}^∞ sin((2n+1)πx/l)/(2n+1)³.

#### Coefficients
Half-range sine series: f(x) = Σ bₙ sin(nπx/l), with bₙ = (2/l) ∫₀^l (lx − x²) sin(nπx/l) dx. Let k = nπ/l (so kl = nπ). Integrate by parts twice:

```text
∫₀^l (lx − x²) sin kx dx = [ −(lx − x²) cos kx/k ]₀^l + (1/k) ∫₀^l (l − 2x) cos kx dx
 = 0 + (1/k) { [ (l − 2x) sin kx/k ]₀^l + (2/k) ∫₀^l sin kx dx }
 = (1/k)(2/k) [ (1 − cos kl)/k ]
 = 2 (1 − (−1)ⁿ)/k³
 = 2 l³ (1 − (−1)ⁿ)/(nπ)³
```

(The boundary terms vanish because sin kl = sin nπ = 0 and lx − x² = 0 at both ends.)

```text
bₙ = (2/l) · 2l³ (1 − (−1)ⁿ)/(n³π³) = 4l² (1 − (−1)ⁿ)/(n³π³)
```

For even n, bₙ = 0. For odd n, put n = 2m + 1: bₙ = 8l²/(π³(2m + 1)³).

#### Result
```text
lx − x² = (8l²/π³) Σ_{m=0}^∞ [1/(2m + 1)³] sin((2m + 1)πx/l),   0 < x < l
```

which is the required series. ∎ (Check at x = l/2: l²/4 = (8l²/π³)(1 − 1/27 + 1/125 − ...) → π³/32 = 1 − 1/3³ + 1/5³ − ..., a known result ✓.)

## Check yourself

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

1. What is being shown?
   A. bn = 0 for even n and 8l^2/(π^3(2m+1)^3) for odd n
   B. bn = 0 for odd n
   C. bn = 8l^2/(nπ) for all n
   D. bn = 4l^2/n^3 for all n
   Answer: A. bn = 4l^2(1 − (−1)^n)/(n^3π^3). (Key point: Strategy)

2. Boundary terms vanish because:
   A. cos nπ = 0
   B. lx − x^2 = 0 at both ends and sin nπ = 0
   C. l = 0
   D. n is even
   Answer: B. Both factors kill the terms. (Key point: Boundary terms)

3. With k = nπ/l, kl equals:
   A. n
   B. π/l
   C. nπ
   D. n^2π^2
   Answer: C. So cos kl = (−1)^n. (Key point: k substitution)

4. For n = 2m + 1, bn equals:
   A. 4l^2/(π^3(2m+1)^3)
   B. 8l^2/(π(2m+1))
   C. 16l^2/(π^3(2m+1)^3)
   D. 8l^2/(π^3(2m+1)^3)
   Answer: D. 1 − (−1)^n = 2, so 4l^2·2/(n^3π^3). (Key point: Odd n)

## Related in Mathematics III

- [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: 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: 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×
- [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: x²](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ahalf-range-sine-series-x2-two) — Mathematics III, chapter 4, asked 9×

## Related in other subjects

- [Fourier series of half-wave rectified sine](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Afs-half-wave-rectified) — Data Communication, chapter 3
- [Sine transform of e^{-x}; ∫x sin mx/(x²+1)](https://www.bctengineer.com/question.md?key=semester_4%2Fapplied_mathematics%3Afourier-sine-transform-x-sin-mx) — Applied Mathematics, chapter 4, asked 4×
- [RMS and average of half-wave rectified sine](https://www.bctengineer.com/question.md?key=semester_1%2Ffundamental_of_electrical_and_electronics_engineering%3Arms-average-half-wave) — Fundamental of Electrical and Electronics Engineering, chapter 2
- [Fourier series of full-wave rectified sine](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Afs-full-wave-rectified) — Data Communication, chapter 3