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

# Laplace transforms of functions te^(−t)sint

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2069 Asar, 5 marks
- Video: [Laplace's transformation of L [ t² e^t sin4t]. Engineering mathematics by shailesh sj Upadhyay sir](https://www.youtube.com/watch?v=zpN5EXxsiYU) — Rohini educational center of mathematics
- 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-t-four
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

Find the Laplace transforms of: (a) t e^(−t) sin t  (b) (e^(at) − cos 6t)/t.

## Short answer

a) t e^{-t} sin t: 2(s+1)/(( (s+1)^2+1)^2. b) Integral from s: ½ ln(s^2+36) - ln(s-a), s>a.

## Answer

#### (a) L[t e^(−t) sin t]
L[sin t] = 1/(s²+1); L[t sin t] = −d/ds[1/(s²+1)] = 2s/(s²+1)².
Shift s → s + 1:

**Answer: 2(s + 1)/((s + 1)² + 1)²**

#### (b) L[(e^(at) − cos 6t)/t]
F(u) = 1/(u − a) − u/(u² + 36).

```text
∫ₛ^∞ F(u) du = [ ln(u − a) − ½ ln(u² + 36) ]ₛ^∞
             = [ ln( (u − a)/√(u² + 36) ) ]ₛ^∞
             = 0 − ln( (s − a)/√(s² + 36) )
```

**Answer: ln( √(s² + 36) / (s − a) ) = ½ ln(s² + 36) − ln(s − a)**, for s > a.

## Check yourself

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

1. After L{t sin t} = 2s/(s^2+1)^2 what shift gives e^(-t)?
   A. s -> s-1
   B. s -> s+1
   C. multiply by e^(-s)
   D. differentiate
   Answer: B. First shifting. (Key point: Shift)

2. What is L{t e^(-t) sin t}?
   A. 2s/((s+1)^2+1)^2
   B. (s+1)/((s+1)^2+1)^2
   C. 2(s+1)/((s+1)^2+1)^2
   D. 2/((s+1)^2+1)^2
   Answer: C. Shifted. (Key point: Result for (a))

3. What is L{(e^(at)-cos 6t)/t}?
   A. ln((s-a)/sqrt(s^2+36))
   B. ln(s^2+36)-ln(s-a)
   C. ln(s-a)+ln sqrt(s^2+36)
   D. ln(sqrt(s^2+36)/(s-a))
   Answer: D. 0 minus ln of ratio. (Key point: Result for (b))

## Close variants set in other exams

- [Laplace transforms of functions te^(−4t)sin3t](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Alaplace-transforms-of-functions-te-4t-two) — Mathematics III, chapter 3, asked 2×

## Related in Mathematics III

- [Laplace transforms of functions te^(−t)sint](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Alaplace-transforms-of-functions-te-t-two) — Mathematics III, chapter 3, asked 10×
- [Laplace transforms of functions te^(−t)sint](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Alaplace-transforms-of-functions-te-t-three) — Mathematics III, chapter 3
- [Laplace transforms of functions te^(−3t)cos2t](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Alaplace-transforms-of-functions-te-3t-two) — Mathematics III, chapter 3, asked 3×
- [Laplace transforms of functions teᵗ](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Alaplace-transforms-of-functions-te) — Mathematics III, chapter 3
- [Laplace transforms of functions t²sint](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Alaplace-transforms-of-functions-t2sint) — Mathematics III, chapter 3
- [Laplace transforms of functions te^(3t)sin3t](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Alaplace-transforms-of-functions-te-3t) — Mathematics III, chapter 3, asked 9×