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

# Half range sine series: e^(ax)

Mathematics III · Chapter 4: Fourier Series

- Asked in: 2083 Baishakh, 5 marks; 2081 Baishakh, 5 marks (asked 2 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-ax
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-4

## Question

Find the half-range Fourier sine series of f(x) = e^(ax) in the interval 0 ≤ x ≤ π.

## Short answer

Using ∫ e^(ax) sin(nx) dx, the half-range sine series on (0,pi) is e^(ax) = (2/pi) sum_{n=1}^infinity [n(1-(-1)^n e^(a pi))/(a^2+n^2)] sin(nx), obtained by integration by parts.

## Answer

Half-range sine series on (0, π): f(x) = Σ bₙ sin nx with

```text
bₙ = (2/π) ∫₀^π e^(ax) sin nx dx
```

Use ∫ e^(ax) sin nx dx = e^(ax)(a sin nx − n cos nx)/(a² + n²):

```text
∫₀^π e^(ax) sin nx dx = [ e^(ax)(a sin nx − n cos nx)/(a² + n²) ]₀^π
                      = [ −n(−1)ⁿ e^(aπ) + n ] / (a² + n²)
                      = n [ 1 − (−1)ⁿ e^(aπ) ] / (a² + n²)
```

```text
bₙ = (2n/π) · [ 1 − (−1)ⁿ e^(aπ) ] / (a² + n²)
```

**Answer:**

```text
e^(ax) = (2/π) Σ_{n=1}^∞ [ n (1 − (−1)ⁿ e^(aπ)) / (a² + n²) ] sin nx,   0 < x < π
```

## Check yourself

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

1. On (0,pi), bn =
   A. (1/pi) integral from -pi to pi
   B. (2/pi) integral from 0 to pi of e^(ax) sin nx dx
   C. (2/pi) integral e^(ax) cos nx
   D. 2 integral from 0 to 1
   Answer: B. Half-range sine. (Key point: Formula)

2. integral e^(ax) sin nx dx =
   A. e^(ax)(a cos nx + n sin nx)/(a^2+n^2)
   B. e^(ax)(n sin nx - a cos nx)/(a^2+n^2)
   C. e^(ax)(a sin nx - n cos nx)/(a^2+n^2)
   D. e^(ax)(a sin nx - n cos nx)/(a^2-n^2)
   Answer: C. Standard. (Key point: Antiderivative)

3. At x = pi, sin n pi = 0 and cos n pi =
   A. 1
   B. -1
   C. n
   D. (-1)^n
   Answer: D. So -n(-1)^n e^(a pi). (Key point: Evaluation)

4. bn =
   A. (2n/pi)[1 - (-1)^n e^(a pi)]/(a^2+n^2)
   B. (2n/pi)[1 + (-1)^n e^(a pi)]/(a^2+n^2)
   C. (2/pi)[1 - (-1)^n e^(a pi)]/(a^2+n^2)
   D. (2n/pi)[1 - (-1)^n e^(a pi)]/(a^2-n^2)
   Answer: A. Combine. (Key point: bn)

## Close variants set in other exams

- [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: 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: 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×

## 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: 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×

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