---
title: "Laplace transforms of functions t²sint"
subject: "Mathematics III"
chapter: 3
key: "semester_3/mathematics_iii:laplace-transforms-of-functions-t2sint"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/laplace-transforms-of-functions-t2sint
site: BCT Engineering
---

# Laplace transforms of functions t²sint

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2068 Chaitra, 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-t2sint
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

Find the Laplace transform of the given function: (i) t² sin t  (ii) cos at sinh at.

## Short answer

Using L[t^2 f]=F''(s) with L[sin t]=1/(s^2+1) gives L[t^2 sin t]=2(3s^2-1)/(s^2+1)^3. Using sinh at = (e^(at)-e^(-at))/2 and first shift gives L[cos at sinh at]=a(s^2-2a^2)/(s^4+4a^4).

## Answer

#### (i) L[t² sin t]
L[sin t] = 1/(s² + 1). Use L[t² f] = F″(s).

```text
F′(s) = −2s/(s² + 1)²
F″(s) = [−2(s²+1)² + 2s·2(s²+1)·2s]/(s²+1)⁴
      = [−2(s²+1) + 8s²]/(s²+1)³
      = (6s² − 2)/(s² + 1)³
```

**Answer: 2(3s² − 1)/(s² + 1)³**

#### (ii) L[cos at sinh at]
sinh at = (e^(at) − e^(−at))/2.

```text
L[cos at sinh at] = ½ [ (s − a)/((s−a)² + a²) − (s + a)/((s+a)² + a²) ]
```

Denominators: (s − a)² + a² = s² − 2as + 2a², (s + a)² + a² = s² + 2as + 2a². Their product is (s² + 2a²)² − 4a²s² = s⁴ + 4a⁴.
Numerator: (s−a)(s² + 2as + 2a²) − (s+a)(s² − 2as + 2a²) = 2a(s² − 2a²).

**Answer: a(s² − 2a²)/(s⁴ + 4a⁴)**

## Check yourself

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

1. Which result is used to find L[t^2 sin t] from F(s) = L[sin t]?
   A. L[t^2 f(t)] = F''(s)
   B. L[t^2 f(t)] = -F'(s)
   C. L[t^2 f(t)] = F(s)/s^2
   D. L[t^2 f(t)] = F(s - 2)
   Answer: A. Multiplying by t^n corresponds to (-1)^n times the n-th derivative of F; for n = 2 the sign is +, so F''(s). (Key point: Multiplication by t^n)

2. What is F'(s) when F(s) = 1/(s^2 + 1)?
   A. 2s/(s^2 + 1)^2
   B. -2s/(s^2 + 1)^2
   C. -1/(s^2 + 1)^2
   D. -2/(s^2 + 1)^2
   Answer: B. Differentiating (s^2+1)^(-1) gives -(s^2+1)^(-2) times 2s. (Key point: First derivative of 1/(s^2+1))

3. What is L[t^2 sin t]?
   A. 2(s^2 - 3)/(s^2 + 1)^3
   B. -2s/(s^2 + 1)^2
   C. 2(3s^2 - 1)/(s^2 + 1)^3
   D. 2(1 - 3s^2)/(s^2 + 1)^3
   Answer: C. Differentiating again and simplifying gives (6s^2 - 2)/(s^2+1)^3. (Key point: Final answer (i))

4. In L[cos at sinh at], the denominators (s-a)^2 + a^2 and (s+a)^2 + a^2 multiply to which expression?
   A. s^4 - 4a^4
   B. (s^2 + a^2)^2
   C. s^4 + a^4
   D. s^4 + 4a^4
   Answer: D. They are s^2 +- 2as + 2a^2, so the product is (s^2+2a^2)^2 - 4a^2 s^2 = s^4 + 4a^4. (Key point: Product of shifted denominators)

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