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

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

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2074 Chaitra, 1+2+2 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-three
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

State the existence theorem for Laplace transform. Obtain the Laplace transform of a) t e^(−t) sin t  b) (e^(−at) − e^(−bt))/t.

## Short answer

Existence: f is piecewise continuous and of exponential order. a) L{e^(−t) sin t}=1/(( s+1)²+1), multiply by t: −d/ds gives 2(s+1)/(s²+2s+2)². b) L{(e^(−at)−e^(−bt))/t} = ∫ₛ^∞ (1/(u+a)−1/(u+b)) du = ln((s+b)/(s+a)).

## Answer

#### Existence theorem
If f(t) is piecewise continuous on every finite interval 0 ≤ t ≤ A and of exponential order (|f(t)| ≤ M e^(kt) for t > T), then L[f(t)] = ∫₀^∞ e^(−st) f(t) dt exists for all s > k.

#### (a) L[t e^(−t) sin t]
Step 1: L[sin t] = 1/(s² + 1).
Step 2 (multiply by t): L[t sin t] = −d/ds [1/(s²+1)] = 2s/(s² + 1)².
Step 3 (first shift, s → s + 1):

```text
L[t e^(−t) sin t] = 2(s + 1) / ((s + 1)² + 1)²
```

**Answer: 2(s + 1)/(s² + 2s + 2)²**

#### (b) L[(e^(−at) − e^(−bt))/t]
Here F(u) = 1/(u + a) − 1/(u + b). Using L[f/t] = ∫ₛ^∞ F(u) du:

```text
∫ₛ^∞ [1/(u+a) − 1/(u+b)] du = [ ln((u + a)/(u + b)) ]ₛ^∞
                            = 0 − ln((s + a)/(s + b))
```

**Answer: ln((s + b)/(s + a))**

## Check yourself

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

1. For which s does L{f} exist under the theorem?
   A. s > k, where |f| <= M e^(kt)
   B. All real s
   C. s < k
   D. Only s = k
   Answer: A. Convergence needs s greater than the growth rate. (Key point: Existence theorem)

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

3. The integral of 1/(u+a) - 1/(u+b) is what?
   A. ln((u+b)/(u+a))
   B. ln(u+a) + ln(u+b)
   C. ln((u+a)/(u+b))
   D. (u+a)/(u+b)
   Answer: C. Difference of logs. (Key point: Antiderivative)

4. What is L{(e^(-at) - e^(-bt))/t}?
   A. ln((s+a)/(s+b))
   B. ln(s+a) - ln(s)
   C. (s+b)/(s+a)
   D. ln((s+b)/(s+a))
   Answer: D. 0 - ln((s+a)/(s+b)) flips the ratio. (Key point: Result for (b))

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