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

# Inverse Laplace transforms

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2072 Kartik, 2+3 marks
- Video: [Inverse Laplace Transform Example using Partial Fractions](https://www.youtube.com/watch?v=c6YnYr8KsSo) — Dr. Trefor Bazett
- 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
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

Find the inverse Laplace transform of the following: a) 1/(s² − 5s + 6)  b) (s + 2)/(s² + 4s + 5)².

## Short answer

1/(s^2-5s+6) = 1/(( s-3)(s-2)) = 1/(s-3)-1/(s-2), inverse is e^(3t)-e^(2t). For (s+2)/(s^2+4s+5)^2, complete square (s+2)^2+1, shift and use L^-1[s/(s^2+1)^2] = (t sin t)/2, yielding ½ t e^(-2t) sin t.

## Answer

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

**Answer: e^(3t) − e^(2t)**

#### b) L⁻¹[(s + 2)/(s² + 4s + 5)²]
Complete the square: s² + 4s + 5 = (s + 2)² + 1. So F(s) = G(s + 2) with G(s) = s/(s² + 1)².

Now L[t sin t] = 2s/(s² + 1)², hence L⁻¹[s/(s²+1)²] = (t sin t)/2. (Or by convolution of cos t and sin t.)

By the first shifting theorem, L⁻¹[G(s + 2)] = e^(−2t) g(t):

**Answer: ½ t e^(−2t) sin t**

## Check yourself

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

1. 1/(s^2 - 5s + 6) =
   A. 1/(s-2) - 1/(s-3)
   B. 1/(s-3) + 1/(s-2)
   C. 1/(s+3) - 1/(s+2)
   D. 1/(s-3) - 1/(s-2)
   Answer: D. A = 1 at s = 3, B = -1 at s = 2. (Key point: Partial fractions (a))

2. L^-1 =
   A. e^(3t) - e^(2t)
   B. e^(2t) - e^(3t)
   C. e^(3t) + e^(2t)
   D. e^(-3t) - e^(-2t)
   Answer: A. Invert each term. (Key point: Result (a))

3. s^2 + 4s + 5 =
   A. (s+2)^2 - 1
   B. (s+2)^2 + 1
   C. (s-2)^2 + 1
   D. (s+4)^2 + 5
   Answer: B. 4 = 2*2 and 5 - 4 = 1. (Key point: Completing square)

4. L^-1[(s+2)/(s^2+4s+5)^2] =
   A. t e^(-2t) sin t
   B. (1/2) t e^(2t) sin t
   C. (1/2) t e^(-2t) sin t
   D. (1/2) t e^(-2t) cos t
   Answer: C. L^-1[s/(s^2+1)^2] = t sin t / 2, shifted by e^(-2t). (Key point: Result (b))

## Close variants set in other exams

- [Inverse Laplace transforms 1/((s−2)(s+2)²)](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ainverse-laplace-transforms-1-s-2-four) — Mathematics III, chapter 3, asked 2×
- [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 (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/(s²−5s+6)](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ainverse-laplace-transforms-1-s2-5s) — Mathematics III, chapter 3, asked 12×
- [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+1)/(s−1))](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Ainverse-laplace-transforms-log-s-1) — 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×

## Related in other subjects

- [R-H root distribution of s⁶+3s⁵+4s⁴+6s³+5s²+3s+2](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arh-s6-3-4-6-5) — Control Systems, chapter 4, asked 5×
- [Bode plot: 20(S+1)/((S²+4S+2)(S²+5S))](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Abode-20s1-s2-4s-2-s2) — Electrical Circuit Theory, chapter 5, asked 4×
- [R-H root distribution of s⁵+3s⁴+5s³+4s²+s+3](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Arh-s5-3-5-4-1) — Control Systems, chapter 4, asked 4×
- [State model from 20(4s+2)/(s³+5s²+8s+2)](https://www.bctengineer.com/question.md?key=semester_3%2Fcontrol_systems%3Ass-from-tf-20-4s2) — Control Systems, chapter 8, asked 2×