---
title: "Laplace transforms of functions te^(3t)sin3t"
subject: "Mathematics III"
chapter: 3
key: "semester_3/mathematics_iii:laplace-transforms-of-functions-te-3t"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/laplace-transforms-of-functions-te-3t
site: BCT Engineering
---

# Laplace transforms of functions te^(3t)sin3t

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2073 Shrawan, 2+3 marks
- Video: [Laplace Transform of te^-2tsin2t || Laplace Transform example](https://www.youtube.com/watch?v=YcfphxmyObY) — 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/laplace-transforms-of-functions-te-3t
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

Find the Laplace transform of: i) t e^(3t) sin 3t  ii) e^(−t) sin t / t.

## Short answer

For t e^(3t) sin3t, start with L[sin3t]=3/(s²+9). Multiply by t: L[t sin3t] = 6s/(s²+9)². Shift s→s-3: 6(s-3)/(( s-3)²+9)². For e^(-t) sin t/t, L[sin t/t] = cot⁻¹ s, shift s→s+1 to get cot⁻¹(s+1).

## Answer

#### i) L[t e^(3t) sin 3t]
L[sin 3t] = 3/(s² + 9).
Multiply by t: L[t sin 3t] = −d/ds [3/(s² + 9)] = 6s/(s² + 9)².
First shift (s → s − 3):

**Answer: 6(s − 3)/((s − 3)² + 9)²**  =  6(s − 3)/(s² − 6s + 18)²

#### ii) L[e^(−t) sin t / t]
First, L[sin t / t] = ∫ₛ^∞ du/(u² + 1) = π/2 − tan⁻¹ s = cot⁻¹ s.
Shift s → s + 1 for the factor e^(−t):

**Answer: cot⁻¹(s + 1) = tan⁻¹(1/(s + 1))**

## Check yourself

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

1. What is L{t sin 3t}?
   A. 3s/(s^2+9)^2
   B. 6/(s^2+9)^2
   C. 3/(s^2+9)^2
   D. 6s/(s^2+9)^2
   Answer: D. -d/ds of 3/(s^2+9). (Key point: L[t sin 3t])

2. How is e^(3t) handled?
   A. Replace s by s - 3
   B. Replace s by s + 3
   C. Multiply by e^(-3s)
   D. Differentiate
   Answer: A. First shifting. (Key point: Shift for e^(3t))

3. What is L{e^(-t) sin t / t}?
   A. cot^-1(s-1)
   B. cot^-1(s+1)
   C. tan^-1(s+1)
   D. 1/((s+1)^2+1)
   Answer: B. Shift s -> s+1 in cot^-1 s. (Key point: Result for (ii))

## Close variants set in other exams

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