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

# Half range sine series: eˣ

Mathematics III · Chapter 4: Fourier Series

- Asked in: 2078 Kartik, 5 marks; 2075 Chaitra, 5 marks; 2069 Asar, 5 marks; 2068 Chaitra, 5 marks (asked 4 times)
- Video: [Half Range sine series f(x)=e^x](https://www.youtube.com/watch?v=BwjvnPvIdWg) — Btech Maths Hub
- 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-e
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-4

## Question

Obtain the half range sine series for f(x) = eˣ in 0 < x < 1.

## Short answer

On (0,1), the half-range sine series coefficient bn = 2∫_0^1 e^x sin(nπx) dx = 2nπ[1-(-1)^n e]/(1+n^2π^2). Hence e^x = 2π Σ_{n=1}^∞ [n(1-(-1)^n e)/(1+n^2π^2)] sin(nπx), 0<x<1.

## Answer

Half-range sine series on (0, l) with l = 1:

```text
f(x) = Σ bₙ sin nπx,    bₙ = 2 ∫₀¹ eˣ sin nπx dx
```

Use ∫ e^(ax) sin(bx) dx = e^(ax)(a sin bx − b cos bx)/(a² + b²) with a = 1, b = nπ:

```text
∫₀¹ eˣ sin nπx dx = [ eˣ (sin nπx − nπ cos nπx) / (1 + n²π²) ]₀¹
                  = [ −nπ e (−1)ⁿ + nπ ] / (1 + n²π²)
                  = nπ [1 − (−1)ⁿ e] / (1 + n²π²)
```

```text
bₙ = 2nπ [1 − (−1)ⁿ e] / (1 + n²π²)
```

**Answer:**

```text
eˣ = 2π Σ_{n=1}^∞ [ n (1 − (−1)ⁿ e) / (1 + n²π²) ] sin nπx,   0 < x < 1
```

First coefficients: b₁ = 2π(1 + e)/(1 + π²) ≈ 2.15, b₂ = 4π(1 − e)/(1 + 4π²) ≈ −0.53.

## Check yourself

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

1. For (0,1), the half-range sine series is
   A. sum bn sin(nx) with bn = (2/pi)...
   B. sum bn cos(n pi x)
   C. sum bn sin(n pi x) with bn = 2 integral from 0 to 1 f sin(n pi x)
   D. sum bn sin(n pi x/2)
   Answer: C. l = 1. (Key point: Interval)

2. integral e^(ax) sin(bx) dx =
   A. e^(ax)(a cos bx + b sin bx)/(a^2+b^2)
   B. e^(ax)(b sin bx - a cos bx)/(a^2+b^2)
   C. e^(ax)(a sin bx + b cos bx)/(a^2-b^2)
   D. e^(ax)(a sin bx - b cos bx)/(a^2+b^2)
   Answer: D. Standard result. (Key point: Integral formula)

3. bn =
   A. 2n pi[1 - (-1)^n e]/(1 + n^2 pi^2)
   B. 2n pi[1 + (-1)^n e]/(1 + n^2 pi^2)
   C. 2[1 - (-1)^n e]/(1 + n^2 pi^2)
   D. 2n pi[1 - (-1)^n e]/(1 - n^2 pi^2)
   Answer: A. Evaluate between 0 and 1. (Key point: bn)

## Close variants set in other exams

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

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

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