---
title: "Inverse Laplace transforms log((s+1)/(s−1))"
subject: "Mathematics III"
chapter: 3
key: "semester_3/mathematics_iii:inverse-laplace-transforms-log-s-1"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/inverse-laplace-transforms-log-s-1
site: BCT Engineering
---

# Inverse Laplace transforms log((s+1)/(s−1))

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2081 Bhadra, 3+2 marks
- Video: [Find the Inverse Laplace Transform of log((s+1)/(s-1))](https://www.youtube.com/watch?v=5YG_xv-anKM) — RK Eduworld (Classic)
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/inverse-laplace-transforms-log-s-1
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

Find the inverse Laplace transform of (i) log((s+1)/(s−1))  (ii) s/((s − 3)(s² + 4)).

## Short answer

(i) 2 sinh t / t = (e^t - e^(-t))/t; (ii) (1/13)(3 e^(3t) - 3 cos 2t + 2 sin 2t).

## Answer

#### (i) L⁻¹{log((s + 1)/(s − 1))}
Let F(s) = ln(s + 1) − ln(s − 1) and f = L⁻¹{F}. Use L{t f(t)} = −F'(s):

```text
F'(s) = 1/(s + 1) − 1/(s − 1)
L{t f(t)} = 1/(s − 1) − 1/(s + 1)
t f(t) = eᵗ − e^(−t) = 2 sinh t
f(t) = 2 sinh t / t
```

#### (ii) L⁻¹{s/((s − 3)(s² + 4))}
Write s/((s − 3)(s² + 4)) = A/(s − 3) + (Bs + C)/(s² + 4). Then s = A(s² + 4) + (Bs + C)(s − 3).

```text
s = 3:  3 = 13A → A = 3/13
s²:     A + B = 0 → B = −3/13
const:  4A − 3C = 0 → C = 4/13
```

```text
F(s) = (3/13)/(s − 3) + (1/13)(−3s + 4)/(s² + 4)
f(t) = (3/13) e^(3t) − (3/13) cos 2t + (2/13) sin 2t
```

**Answers:** (i) 2 sinh t / t = (eᵗ − e^(−t))/t; (ii) (1/13)(3e^(3t) − 3cos 2t + 2 sin 2t).

## Check yourself

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

1. For F = ln(s+1) - ln(s-1), what is L{t f(t)} = -F'(s)?
   A. 1/(s+1) - 1/(s-1)
   B. 1/(s-1) + 1/(s+1)
   C. 1/(s-1) - 1/(s+1)
   D. ln((s+1)/(s-1))
   Answer: C. F' = 1/(s+1) - 1/(s-1); negating gives 1/(s-1) - 1/(s+1). (Key point: Sign convention)

2. What is L^-1{log((s+1)/(s-1))}?
   A. 2 sinh t
   B. 2 cosh t / t
   C. (e^t + e^(-t))/t
   D. 2 sinh t / t
   Answer: D. t f(t) = e^t - e^(-t) = 2 sinh t, so f = 2 sinh t / t. (Key point: Result for part (i))

3. s = A(s^2+4) + (Bs+C)(s-3). Put s = 3 for A.
   A. 3/13
   B. 1/13
   C. 3/5
   D. 4/13
   Answer: A. 3 = A(9+4) = 13A, so A = 3/13. (Key point: Finding A in part (ii))

4. Comparing constants gives 4A - 3C = 0. What is C?
   A. 3/13
   B. 4/13
   C. 12/13
   D. -4/13
   Answer: B. 3C = 4A = 12/13, so C = 4/13. (Key point: Constant term for C)

## Close variants set in other exams

- [Inverse Laplace transforms 1/(s²(s²+1))and](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ainverse-laplace-transforms-1-s2-s2) — Mathematics III, chapter 3, asked 14×
- [Inverse Laplace transforms log(s/(s+1))](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ainverse-laplace-transforms-log-s-s) — Mathematics III, chapter 3, asked 14×
- [Inverse Laplace transforms (s+2)/(s²−4s+13)](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ainverse-laplace-transforms-s-2-s2) — Mathematics III, chapter 3, asked 14×
- [Inverse Laplace transforms 1/((s−2)(s+4))](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ainverse-laplace-transforms-1-s-2-three) — Mathematics III, chapter 3, asked 14×
- [Inverse Laplace transforms 1/(s²(s²+a²))](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ainverse-laplace-transforms-1-s2-s2-two) — Mathematics III, chapter 3, asked 14×
- [Inverse Laplace transforms s/((s−3)(s²+4))](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ainverse-laplace-transforms-s-s-3) — Mathematics III, chapter 3, asked 14×
- [Inverse Laplace transforms 1/(s(s−1))](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ainverse-laplace-transforms-1-s-s) — Mathematics III, chapter 3, asked 23×
- [Inverse Laplace transforms s²/(s−1)³](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ainverse-laplace-transforms-s2-s-1) — Mathematics III, chapter 3, asked 23×
- [Inverse Laplace transforms (s+3)/(s²+6s+13)²](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ainverse-laplace-transforms-s-3-s2) — Mathematics III, chapter 3, asked 23×
- [Inverse Laplace transforms 1/(s²−3s+2)](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ainverse-laplace-transforms-1-s2-3s) — Mathematics III, chapter 3, asked 23×
- [Inverse Laplace transforms 1/(s(s+1))](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ainverse-laplace-transforms-1-s-s-two) — Mathematics III, chapter 3, asked 23×
- [Inverse Laplace transforms (2s+3)/(s²+5s−6)](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ainverse-laplace-transforms-2s-3-s2) — Mathematics III, chapter 3, asked 23×
- [Inverse Laplace transforms (s+2)³/s⁴](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ainverse-laplace-transforms-s-2-3) — Mathematics III, chapter 3, asked 23×

## Related in Mathematics III

- [Inverse Laplace transforms e^(−πs)(s+1)/(s²+2s+2)](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ainverse-laplace-transforms-e-s-s) — Mathematics III, chapter 3, asked 12×
- [Inverse Laplace transforms 1/(s(s+2))](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ainverse-laplace-transforms-1-s-s-three) — Mathematics III, chapter 3, asked 12×
- [Inverse Laplace transforms 1/((s−2)(s+4))](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ainverse-laplace-transforms-1-s-2) — Mathematics III, chapter 3, asked 10×
- [Inverse Laplace transforms (s+2)/(s+1)⁴](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ainverse-laplace-transforms-s-2-s) — Mathematics III, chapter 3, asked 10×