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

# Laplace transforms of functions coshatsinat

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2067 Asar
- Video: [Laplace transform: cosh(at) and sinh(at)](https://www.youtube.com/watch?v=4snzBywrY4E) — Papa Flammy
- 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-coshatsinat
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

Find the Laplace transform of: a) cosh at sin at  b) (cos 2t − cos 3t)/t.

## Short answer

Using cosh at = (e^(at)+e^(-at))/2 gives L[cosh at sin at] = a(s^2+2a^2)/(s^4+4a^4). Using integration property gives L[(cos 2t - cos 3t)/t] = (1/2) ln((s^2+9)/(s^2+4)).

## Answer

#### a) L[cosh at sin at]
cosh at = (e^(at) + e^(−at))/2 and L[sin at] = a/(s² + a²). By the first shift,

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

Denominators: (s − a)² + a² = s² − 2as + 2a² and (s + a)² + a² = s² + 2as + 2a²; the product is (s² + 2a²)² − 4a²s² = s⁴ + 4a⁴. The numerator is a[(s² + 2as + 2a²) + (s² − 2as + 2a²)] = 2a(s² + 2a²).

**Answer: a(s² + 2a²)/(s⁴ + 4a⁴)**

#### b) L[(cos 2t − cos 3t)/t]
```text
∫ₛ^∞ [u/(u²+4) − u/(u²+9)] du = ½ [ ln((u²+4)/(u²+9)) ]ₛ^∞
                              = −½ ln((s²+4)/(s²+9))
```

**Answer: ½ ln((s² + 9)/(s² + 4))**

## Check yourself

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

1. How is cosh at sin at transformed in part (a)?
   A. Write cosh at as (e^(at)+e^(-at))/2 and use the first shift on sin at
   B. Use the integral of F(u)
   C. Differentiate L[sin at]
   D. Use the second shifting theorem
   Answer: A. Each exponential shifts s by -+a. (Key point: Exponential form)

2. What is L[cosh at sin at]?
   A. a(s^2 - 2a^2)/(s^4 + 4a^4)
   B. a(s^2 + 2a^2)/(s^4 + 4a^4)
   C. a(s^2 + 2a^2)/(s^4 - 4a^4)
   D. 2a(s^2 + a^2)/(s^4 + 4a^4)
   Answer: B. The numerator is 2a(s^2+2a^2) before the 1/2. (Key point: Result (a))

3. Part (b), (cos 2t - cos 3t)/t, uses
   A. Derivative of F(s)
   B. F(s)/s
   C. Integral of F(u) from s to infinity
   D. Shift of s
   Answer: C. L[f/t] = integral of F from s to infinity. (Key point: Division by t)

4. What is L[(cos 2t - cos 3t)/t]?
   A. (1/2) ln((s^2+4)/(s^2+9))
   B. ln((s^2+9)/(s^2+4))
   C. (1/2) ln((s+3)/(s+2))
   D. (1/2) ln((s^2+9)/(s^2+4))
   Answer: D. The integral gives -(1/2) ln((s^2+4)/(s^2+9)). (Key point: Result (b))

## Close variants set in other exams

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