---
title: "Inverse Laplace transforms tan⁻¹(1/s)"
subject: "Mathematics III"
chapter: 3
key: "semester_3/mathematics_iii:inverse-laplace-transforms-taninverse-1-s"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/inverse-laplace-transforms-taninverse-1-s
site: BCT Engineering
---

# Inverse Laplace transforms tan⁻¹(1/s)

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2080 Bhadra, 2+3 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/inverse-laplace-transforms-taninverse-1-s
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

Find the inverse Laplace transform of a) tan⁻¹(1/s)  b) 1/(s(s − 4)²).

## Short answer

a) sin t / t; b) 1/16 (1 - e^(4t)) + (t/4) e^(4t).

## Answer

#### a) L⁻¹{tan⁻¹(1/s)}
Let F(s) = tan⁻¹(1/s) and f = L⁻¹{F}. Using L{t f(t)} = −F'(s):

```text
F'(s) = −1/(1 + s²)
L{t f(t)} = 1/(s² + 1)  → t f(t) = sin t
f(t) = sin t / t
```

#### b) L⁻¹{1/(s(s − 4)²)}
Partial fractions: 1/(s(s − 4)²) = A/s + B/(s − 4) + C/(s − 4)².

```text
A = 1/16        (s = 0)
C = 1/4         (s = 4)
A + B = 0 (large s behaviour) → B = −1/16
L⁻¹ = 1/16 − (1/16) e^(4t) + (1/4) t e^(4t)
```

**Answers:** a) sin t / t; b) (1/16)(1 − e^(4t)) + (t/4) e^(4t).

## Check yourself

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

1. For F(s) = tan^-1(1/s), what is F'(s)?
   A. 1/(1 + s^2)
   B. -1/(1 + s^2)
   C. -1/s^2
   D. -s/(1 + s^2)
   Answer: B. d/ds tan^-1(1/s) = -1/(1+s^2), the same as for cot^-1 s. (Key point: Derivative of F)

2. What is L^-1{tan^-1(1/s)}?
   A. sin t
   B. cos t / t
   C. sin t / t
   D. t sin t
   Answer: C. L{t f} = 1/(s^2+1) gives t f = sin t, so f = sin t / t. (Key point: Result for part (a))

3. 1/(s(s-4)^2) is decomposed as?
   A. A/s + B/(s-4)^2
   B. A/s + B/(s-4)
   C. (As+B)/(s-4)^2
   D. A/s + B/(s-4) + C/(s-4)^2
   Answer: D. A repeated factor needs terms for each power up to 2. (Key point: Form of partial fractions)

4. Multiply by (s-4)^2 and set s = 4. What is C?
   A. 1/4
   B. 1/16
   C. 1/2
   D. 4
   Answer: A. C = 1/s at s = 4 = 1/4. (Key point: Value of C)

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