---
title: "Laplace transforms of functions e^(2t)cos²2t"
subject: "Mathematics III"
chapter: 3
key: "semester_3/mathematics_iii:laplace-transforms-of-functions-e-2t"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/laplace-transforms-of-functions-e-2t
site: BCT Engineering
---

# Laplace transforms of functions e^(2t)cos²2t

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2075 Asoj, 1+2+2 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-e-2t
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

State the conditions for existence of Laplace transform. Obtain the Laplace transform of: (i) e^(2t) cos² 2t  (ii) (cos 2t − cos 3t)/t.

## Short answer

Existence: f must be piecewise continuous on finite intervals and of exponential order. i) cos²2t = (1+cos4t)/2, shift s→s−2: ½[1/(s−2) + (s−2)/(( s−2)²+16)]. ii) L{(cos2t−cos3t)/t} = ∫ₛ^∞ (u/(u²+4)−u/(u²+9)) du = ½ ln((s²+9)/(s²+4)).

## Answer

#### Conditions for existence
The Laplace transform L[f(t)] = ∫₀^∞ e^(−st) f(t) dt exists if
1. f(t) is piecewise continuous on every finite interval [0, A], and
2. f(t) is of exponential order, that is |f(t)| ≤ M e^(kt) for all t > T, for some constants M, k, T.

These conditions are sufficient (not necessary); then L[f] exists for s > k.

#### (i) L[e^(2t) cos² 2t]
cos² 2t = (1 + cos 4t)/2, so

```text
L[cos² 2t] = ½ [1/s + s/(s² + 16)]
```

By the first shifting theorem, replace s by (s − 2):

```text
L[e^(2t) cos² 2t] = ½ [ 1/(s − 2) + (s − 2)/((s − 2)² + 16) ]
```

**Answer: ½ [1/(s − 2) + (s − 2)/((s − 2)² + 16)]**

#### (ii) L[(cos 2t − cos 3t)/t]
Use L[f(t)/t] = ∫ₛ^∞ F(u) du with F(u) = u/(u² + 4) − u/(u² + 9).

```text
∫ₛ^∞ [u/(u²+4) − u/(u²+9)] du = ½ [ ln((u²+4)/(u²+9)) ]ₛ^∞
                              = ½ [ 0 − 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. Which conditions guarantee that L{f(t)} exists for s > k?
   A. f is differentiable everywhere and bounded
   B. f is piecewise continuous on every finite interval and of exponential order |f(t)| <= M e^(kt)
   C. f is periodic and continuous at t = 0
   D. f is continuous with f(0) = 0 and f(t) -> 0 as t -> infinity
   Answer: B. The answer states these two conditions: piecewise continuity on each [0, T] and exponential order. They are sufficient, and the transform then exists for s > k. (Key point: Conditions for the Laplace transform to exist)

2. cos^2 2t equals what?
   A. (1 + cos 2t)/2
   B. (1 - cos 4t)/2
   C. (1 + cos 4t)/2
   D. cos 4t
   Answer: C. Half-angle identity. (Key point: Identity)

3. How is e^(2t) handled in (i)?
   A. Replace s by s + 2
   B. Multiply by e^(-2s)
   C. Differentiate
   D. Replace s by s - 2
   Answer: D. First shifting theorem. (Key point: Shift direction)

4. What is L{(cos 2t - cos 3t)/t}?
   A. (1/2) ln((s^2+9)/(s^2+4))
   B. (1/2) ln((s^2+4)/(s^2+9))
   C. ln((s^2+9)/(s^2+4))
   D. (1/2)(s/(s^2+4) - s/(s^2+9))
   Answer: A. Integrate from s to infinity. (Key point: Result for (ii))

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