---
title: "Laplace transforms of functions t³e^(−3t)and"
subject: "Mathematics III"
chapter: 3
key: "semester_3/mathematics_iii:laplace-transforms-of-functions-t3e-3t"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/laplace-transforms-of-functions-t3e-3t
site: BCT Engineering
---

# Laplace transforms of functions t³e^(−3t)and

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2082 Baishakh, 2+3 marks
- Video: [8. Laplace Transforms | Multiplication by t^n | Problem#5 | Complete Concept](https://www.youtube.com/watch?v=bTsZO1W-dVE) — MKS TUTORIALS by Manoj Sir
- 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-t3e-3t
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

Find the Laplace transform of (a) t³e^(−3t) and (b) (e^(at) − cos 6t)/t.

## Short answer

By shifting theorem, L{t³e^(−3t)}=6/(s+3)⁴. Use L{f(t)/t}=∫ₛ^∞ F(u) du with F(s)=1/(s−a)−s/(s²+36) to get L{(e^(at)−cos6t)/t}=ln(√(s²+36)/(s−a)).

## Answer

#### (a) L{t³ e^(−3t)}
By the first shifting theorem, L{e^(−3t) f(t)} = F(s + 3), with L{t³} = 3!/s⁴ = 6/s⁴:

```text
L{t³ e^(−3t)} = 6/(s + 3)⁴
```

#### (b) L{(e^(at) − cos 6t)/t}
Use L{f(t)/t} = ∫ₛ^∞ F(u) du, with F(s) = 1/(s − a) − s/(s² + 36). (As t → 0, f(t)/t → a, finite.)

```text
∫ₛ^∞ [1/(u − a) − u/(u² + 36)] du = [ln (u − a) − ½ ln(u² + 36)]ₛ^∞
 = [ln ((u − a)/√(u² + 36))]ₛ^∞
 = 0 − ln ((s − a)/√(s² + 36))
 = ln (√(s² + 36)/(s − a))        (s > a)
```

**Answers:** (a) 6/(s + 3)⁴; (b) ln(√(s² + 36)/(s − a)) = ½ ln(s² + 36) − ln(s − a).

## Check yourself

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

1. Multiplying f(t) by e^(-3t) does what to F(s)?
   A. Replaces s by s - 3
   B. Multiplies F by e^(-3s)
   C. Differentiates F three times
   D. Replaces s by s + 3
   Answer: D. First shifting theorem: L{e^(-at) f} = F(s+a). (Key point: Effect of e^(-3t))

2. What is L{t^3 e^(-3t)}?
   A. 6/(s+3)^4
   B. 6/(s-3)^4
   C. 3/(s+3)^4
   D. 6/(s+3)^3
   Answer: A. L{t^3} = 6/s^4, then s -> s+3. (Key point: Part (a))

3. Which condition is needed to use L{f/t} = integral of F from s to infinity?
   A. f must be periodic
   B. f(t)/t has a finite limit as t -> 0
   C. f(0) must equal 1
   D. f(t)/t must tend to infinity as t -> 0
   Answer: B. Here (e^(at) - cos 6t)/t -> a as t -> 0, so the formula applies. (Key point: Validity of dividing by t)

4. What is L{(e^(at) - cos 6t)/t}?
   A. ln( (s-a)/sqrt(s^2+36) )
   B. ln( sqrt(s^2+36) (s-a) )
   C. ln( sqrt(s^2+36)/(s-a) )
   D. (1/2) ln(s^2+36) + ln(s-a)
   Answer: C. The integral equals 0 - ln((s-a)/sqrt(s^2+36)), which flips the fraction. (Key point: Result for part (b))

## Close variants set in other exams

- [Laplace transforms of functions t²cosat](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Alaplace-transforms-of-functions-t2cosat-three) — Mathematics III, chapter 3, asked 7×
- [Laplace transforms of functions t²cosat](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Alaplace-transforms-of-functions-t2cosat) — Mathematics III, chapter 3, asked 4×
- [Laplace transforms of functions te^(−t)sint](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Alaplace-transforms-of-functions-te-t-two) — Mathematics III, chapter 3, asked 10×
- [Laplace transforms of functions t²cosat](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Alaplace-transforms-of-functions-t2cosat-two) — Mathematics III, chapter 3, asked 10×
- [Laplace transforms of functions f(t)=(1−cost)/t](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Alaplace-transforms-of-functions-f-t) — Mathematics III, chapter 3, asked 10×
- [Laplace transforms of functions sinhatcosbt](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Alaplace-transforms-of-functions-sinhatcosbt) — Mathematics III, chapter 3, asked 4×

## Related in Mathematics III

- [Laplace transforms of functions te^(−3t)cos2t](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Alaplace-transforms-of-functions-te-3t-two) — Mathematics III, chapter 3, asked 3×
- [Laplace transforms of functions te^(3t)sin3t](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Alaplace-transforms-of-functions-te-3t) — Mathematics III, chapter 3, asked 9×
- [Laplace transforms of functions tsin²3t](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Alaplace-transforms-of-functions-tsin23t) — Mathematics III, chapter 3, asked 9×
- [Laplace transforms of functions e^(2t)cos²2t](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Alaplace-transforms-of-functions-e-2t) — Mathematics III, chapter 3, asked 4×
- [Laplace transforms of functions cos³2t](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Alaplace-transforms-of-functions-cos32t) — Mathematics III, chapter 3, asked 4×
- [Laplace transforms of functions te^(−4t)sin3t](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Alaplace-transforms-of-functions-te-4t-two) — Mathematics III, chapter 3, asked 2×