---
title: "Inverse Laplace transforms e^(−πs)(s+1)/(s²+2s+2)"
subject: "Mathematics III"
chapter: 3
key: "semester_3/mathematics_iii:inverse-laplace-transforms-e-s-s"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/inverse-laplace-transforms-e-s-s
site: BCT Engineering
---

# Inverse Laplace transforms e^(−πs)(s+1)/(s²+2s+2)

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2076 Asoj, 2+3 marks
- Video: [Inverse Laplace Transform Examples || First Shifting Property of Inverse Laplace Transform ||](https://www.youtube.com/watch?v=IVJjm5FE4x8) — Let's Unlock Maths
- 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-e-s-s
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

Find the inverse Laplace transform of: i) e^(−πs)(s + 1)/(s² + 2s + 2)  ii) tan⁻¹(2/s).

## Short answer

i) Complete square to (s+1)²+1, shift, then apply second shifting theorem: e^(−(t−π)) cos(t−π) u(t−π) = −e^(π−t) cos t u(t−π). ii) Differentiate tan⁻¹(2/s) to −2/(s²+4); using L{t f(t)} = −F'(s) gives f(t) = sin 2t / t.

## Answer

#### i) L⁻¹{e^(−πs)(s + 1)/(s² + 2s + 2)}
Complete the square: s² + 2s + 2 = (s + 1)² + 1. So (s + 1)/((s + 1)² + 1) is the shift of s/(s² + 1):

```text
L⁻¹{(s + 1)/((s + 1)² + 1)} = e^(−t) cos t
```

By the second shifting theorem with a = π:

```text
L⁻¹ = e^(−(t−π)) cos(t − π) u(t − π) = −e^(π−t) cos t · u(t − π)
```

where u is the unit step function (cos(t − π) = −cos t).

#### ii) L⁻¹{tan⁻¹(2/s)}
Let F(s) = tan⁻¹(2/s). Then F'(s) = −2/(s² + 4), and using L{t f(t)} = −F'(s):

```text
L{t f(t)} = 2/(s² + 4)  → t f(t) = sin 2t
f(t) = sin 2t / t
```

**Answers:** i) e^(−(t−π)) cos(t − π) u(t − π) = −e^(π−t) cos t u(t − π); ii) sin 2t / t.

## Check yourself

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

1. What is L^-1{(s+1)/((s+1)^2+1)}?
   A. e^(-t) sin t
   B. e^(t) cos t
   C. cos t
   D. e^(-t) cos t
   Answer: D. Shift of s/(s^2+1) with s -> s+1. (Key point: Base inverse)

2. Which theorem handles e^(-pi s)?
   A. Second shifting: f(t - pi) u(t - pi)
   B. First shifting: e^(-pi t) f(t)
   C. Convolution
   D. Derivative theorem
   Answer: A. e^(-as)F(s) is the transform of f(t-a)u(t-a). (Key point: Effect of e^(-pi s))

3. cos(t - pi) equals what?
   A. cos t
   B. -cos t
   C. sin t
   D. -sin t
   Answer: B. A shift by pi flips the sign. (Key point: Simplifying cos(t - pi))

4. What is L^-1{tan^-1(2/s)}?
   A. sin 2t
   B. 2 sin 2t / t
   C. sin 2t / t
   D. cos 2t / t
   Answer: C. L{t f} = -F' = 2/(s^2+4). (Key point: Part (ii))

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