---
title: "Laplace transforms of functions t²sin2t"
subject: "Mathematics III"
chapter: 3
key: "semester_3/mathematics_iii:laplace-transforms-of-functions-t2sin2t-two"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/laplace-transforms-of-functions-t2sin2t-two
site: BCT Engineering
---

# Laplace transforms of functions t²sin2t

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2070 Chaitra, 2.5+2.5 marks
- Video: [Laplace transform of t^2 sinat, Laplace transform](https://www.youtube.com/watch?v=iUZsaYhkjMQ) — Excellence Academy
- 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-t2sin2t-two
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

Find the Laplace transform of (a) t² sin 2t  and (b) (1 − eᵗ)/t.

## Short answer

a) Using L[t^2 f]=F''(s): 4(3s^2-4)/(s^2+4)^3. b) Integration of transform: L[(1-e^t)/t] = ln((s-1)/s), s>1.

## Answer

#### (a) L[t² sin 2t]
L[sin 2t] = 2/(s² + 4). Use L[t² f] = F″(s).

```text
F′(s) = −4s/(s² + 4)²
F″(s) = [−4(s²+4)² + 4s·2(s²+4)·2s]/(s²+4)⁴
      = [−4(s²+4) + 16s²]/(s²+4)³
      = (12s² − 16)/(s²+4)³
```

**Answer: 4(3s² − 4)/(s² + 4)³**

#### (b) L[(1 − eᵗ)/t]
F(u) = 1/u − 1/(u − 1) for u > 1.

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

**Answer: ln((s − 1)/s)**, valid for s > 1.

## Check yourself

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

1. L{t^2 f} = ?
   A. -F''(s)
   B. F''(s)
   C. F'(s)
   D. integral of F
   Answer: B. Even power. (Key point: Formula)

2. What is L{t^2 sin 2t}?
   A. 4(3s^2+4)/(s^2+4)^3
   B. -4(3s^2-4)/(s^2+4)^3
   C. 4(3s^2-4)/(s^2+4)^3
   D. 4s/(s^2+4)^2
   Answer: C. F'' computed. (Key point: Result for (a))

3. F(u) for (1 - e^t)/t?
   A. 1/u + 1/(u-1)
   B. 1/(u-1) - 1/u
   C. u - (u-1)
   D. 1/u - 1/(u-1)
   Answer: D. L{1} - L{e^t}. (Key point: Integrand for (b))

4. What is the transform?
   A. ln((s-1)/s)
   B. ln(s/(s-1))
   C. ln(s(s-1))
   D. ln s - 1
   Answer: A. Valid for s>1. (Key point: Result for (b))

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