---
title: "Inverse Laplace transforms 1/(s²−5s+6)"
subject: "Mathematics III"
chapter: 3
key: "semester_3/mathematics_iii:inverse-laplace-transforms-1-s2-5s"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/inverse-laplace-transforms-1-s2-5s
site: BCT Engineering
---

# Inverse Laplace transforms 1/(s²−5s+6)

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2074 Chaitra, 2.5+2.5 marks
- Video: [13. Inverse Laplace Transforms | Problem#1 | Very Important](https://www.youtube.com/watch?v=zczBU4fsEkY) — MKS TUTORIALS by Manoj Sir
- 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-1-s2-5s
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

Find the inverse Laplace transform of: a) 1/(s² − 5s + 6)  b) tan⁻¹(2/s).

## Short answer

Factor denominator: 1/(( s-2)(s-3)) = 1/(s-3) - 1/(s-2); inverse is e^(3t) - e^(2t). For tan⁻¹(2/s), differentiate: -F'(s) = 2/(s²+4) so t f(t) = sin 2t, giving f(t) = sin 2t / t.

## Answer

#### a) L⁻¹[1/(s² − 5s + 6)]
Factorise: s² − 5s + 6 = (s − 2)(s − 3).

```text
1/((s−2)(s−3)) = 1/(s − 3) − 1/(s − 2)
```

**Answer: e^(3t) − e^(2t)**

#### b) L⁻¹[tan⁻¹(2/s)]
Let F(s) = tan⁻¹(2/s). Then

```text
F′(s) = [1/(1 + 4/s²)] (−2/s²) = −2/(s² + 4)
```

Since L[t f(t)] = −F′(s) = 2/(s² + 4), we get t f(t) = sin 2t.

**Answer: f(t) = sin 2t / t**

## Check yourself

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

1. s^2 - 5s + 6 factors as?
   A. (s-2)(s-3)
   B. (s+2)(s+3)
   C. (s-1)(s-6)
   D. (s-2)(s+3)
   Answer: A. Sum 5, product 6. (Key point: Factorise)

2. What is the inverse?
   A. e^(2t) - e^(3t)
   B. e^(3t) - e^(2t)
   C. e^(2t) + e^(3t)
   D. e^(-2t) - e^(-3t)
   Answer: B. 1/((s-2)(s-3)) = 1/(s-3) - 1/(s-2). (Key point: Result for (a))

3. F' for F = tan^-1(2/s)?
   A. 2/(s^2+4)
   B. -2/(s^2-4)
   C. -2/(s^2+4)
   D. -1/(s^2+4)
   Answer: C. Chain rule gives -2/(s^2+4). (Key point: Derivative)

4. What is the inverse?
   A. sin 2t
   B. cos 2t / t
   C. 2 sin 2t / t
   D. sin 2t / t
   Answer: D. t f = sin 2t. (Key point: Result for (b))

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