---
title: "Laplace transforms of functions te^(−4t)sin3t"
subject: "Mathematics III"
chapter: 3
key: "semester_3/mathematics_iii:laplace-transforms-of-functions-te-4t-two"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/laplace-transforms-of-functions-te-4t-two
site: BCT Engineering
---

# Laplace transforms of functions te^(−4t)sin3t

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2069 Chaitra, 2.5+2.5 marks
- 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-4t-two
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

Find the Laplace transforms of the following functions: a) t e^(−4t) sin 3t  b) (cos at − cos bt)/t.

## Short answer

a) t e^{-4t} sin 3t: 6(s+4)/( ( (s+4)^2+9 )^2. b) (cos at - cos bt)/t: ½ ln((s^2+b^2)/(s^2+a^2)).

## Answer

#### a) L[t e^(−4t) sin 3t]
L[sin 3t] = 3/(s² + 9).
L[t sin 3t] = −d/ds [3/(s²+9)] = 6s/(s² + 9)².
Shift s → s + 4:

**Answer: 6(s + 4)/((s + 4)² + 9)²**  =  6(s + 4)/(s² + 8s + 25)²

#### b) L[(cos at − cos bt)/t]
F(u) = u/(u² + a²) − u/(u² + b²).

```text
∫ₛ^∞ F(u) du = ½ [ ln((u² + a²)/(u² + b²)) ]ₛ^∞
             = ½ [ 0 − ln((s² + a²)/(s² + b²)) ]
```

**Answer: ½ ln((s² + b²)/(s² + a²))**

## 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. 6s/(s^2+9)^2
   B. 3s/(s^2+9)^2
   C. 6/(s^2+9)^2
   D. 3/(s^2+9)^2
   Answer: A. -d/ds 3/(s^2+9). (Key point: L[t sin 3t])

2. e^(-4t) corresponds to?
   A. s -> s-4
   B. s -> s+4
   C. multiply by e^(-4s)
   D. divide by s+4
   Answer: B. First shifting. (Key point: Shift)

3. What is L{(cos at - cos bt)/t}?
   A. (1/2) ln((s^2+a^2)/(s^2+b^2))
   B. ln((s^2+b^2)/(s^2+a^2))
   C. (1/2) ln((s^2+b^2)/(s^2+a^2))
   D. (1/2)(s/(s^2+a^2)-s/(s^2+b^2))
   Answer: C. Flip from minus sign. (Key point: Result for (b))

## Close variants set in other exams

- [Laplace transforms of functions te^(−t)sint](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Alaplace-transforms-of-functions-te-t-four) — Mathematics III, chapter 3, asked 2×

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