---
title: "Laplace transforms of functions t²cosat"
subject: "Mathematics III"
chapter: 3
key: "semester_3/mathematics_iii:laplace-transforms-of-functions-t2cosat-two"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/laplace-transforms-of-functions-t2cosat-two
site: BCT Engineering
---

# Laplace transforms of functions t²cosat

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2074 Asoj, 3+2 marks
- Video: [Problem 1 Based on Multiplication By 't' Property - Laplace Transform - Engineering Mathematics 3](https://www.youtube.com/watch?v=B_f5rJ-Bues) — Ekeeda
- 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-t2cosat-two
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

Find the Laplace transform of: i) t² cos at  ii) sin t / t.

## Short answer

For t² cos at, use L[t² f(t)] = F''(s) with F(s)=s/(s²+a²); result: 2s(s²-3a²)/(s²+a²)³. For sin t / t, use L[ f(t)/t] = ∫_s^∞ F(u) du; L[sin t]=1/(s²+1), integral gives π/2 - tan⁻¹ s = cot⁻¹ s.

## Answer

#### i) L[t² cos at]
Use L[t² f(t)] = d²F/ds² with F(s) = L[cos at] = s/(s² + a²).

```text
F′(s)  = (a² − s²)/(s² + a²)²
F″(s)  = [−2s(s² + a²)² − (a² − s²)·2(s² + a²)·2s] / (s² + a²)⁴
       = [−2s(s² + a²) − 4s(a² − s²)] / (s² + a²)³
       = (2s³ − 6a²s) / (s² + a²)³
```

**Answer: 2s(s² − 3a²)/(s² + a²)³**

#### ii) L[sin t / t]
L[sin t] = 1/(s² + 1). By L[f/t] = ∫ₛ^∞ F(u) du:

```text
∫ₛ^∞ du/(u² + 1) = [tan⁻¹ u]ₛ^∞ = π/2 − tan⁻¹ s
```

**Answer: cot⁻¹ s = tan⁻¹(1/s)**

## Check yourself

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

1. L{t^2 f} equals?
   A. F''(s)
   B. -F''(s)
   C. F'(s)
   D. integral of F from s to infinity
   Answer: A. Sign is (+) for even n. (Key point: Formula for t^2)

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

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

## Close variants set in other exams

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