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

# Laplace transforms of functions sinhatcosbt

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2079 Baishakh, 5 marks
- 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-sinhatcosbt
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

Find the Laplace transform of: (i) sinh at cos bt  (ii) (e^(−at) − e^(−bt))/t.

## Short answer

(i) sinh(at)cos(bt) transforms to a(s^2 - a^2 - b^2) / [((s-a)^2+b^2)((s+a)^2+b^2)]. (ii) (e^(-at)-e^(-bt))/t transforms to ln((s+b)/(s+a)).

## Answer

#### (i) L[sinh at cos bt]
sinh at = (e^(at) − e^(−at))/2 and L[cos bt] = s/(s² + b²). First shifting theorem: L[e^(at) cos bt] = (s − a)/((s − a)² + b²).

```text
L[sinh at cos bt] = ½ [ (s − a)/((s − a)² + b²) − (s + a)/((s + a)² + b²) ]
```

The numerator after combining: (s − a)((s + a)² + b²) − (s + a)((s − a)² + b²) = 2a(s² − a² − b²).

**Answer: a(s² − a² − b²) / [ ((s − a)² + b²)((s + a)² + b²) ]**

#### (ii) L[(e^(−at) − e^(−bt))/t]
```text
∫ₛ^∞ [1/(u + a) − 1/(u + b)] du = [ ln((u + a)/(u + b)) ]ₛ^∞ = −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. sinh at cos bt is transformed by
   A. using the integral theorem
   B. differentiating L[cos bt]
   C. writing sinh at = (e^(at) - e^(-at))/2 and using the first shift
   D. convolution
   Answer: C. Shifts s by -+a. (Key point: Approach (i))

2. The combined numerator equals
   A. 2a(s^2 + a^2 + b^2)
   B. 2a(s^2 - a^2 + b^2)
   C. 2s(s^2 - a^2 - b^2)
   D. 2a(s^2 - a^2 - b^2)
   Answer: D. Expand and simplify. (Key point: Numerator)

3. (e^(-at) - e^(-bt))/t uses
   A. integral of F(u) from s to infinity
   B. -F'(s)
   C. F(s)/s
   D. shift
   Answer: A. Division by t. (Key point: Division by t)

4. L =
   A. ln((s+a)/(s+b))
   B. ln((s+b)/(s+a))
   C. ln((s-b)/(s-a))
   D. (1/2) ln((s+b)/(s+a))
   Answer: B. -ln((s+a)/(s+b)). (Key point: Result (ii))

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