Q.49 The Laplace transform of the function f(t) = t2 + 2t + 1 is
(A) 1/s3 + 3/s2 + 1/s
(B) 4/s3 + 4/s2 + 1/s
(C) 2/s3 + 2/s2 + 1/s
(D) 2/s3 + 3/s2 + 1/s
Introduction: Laplace Transform of t² + 2t + 1
The Laplace transform simplifies calculus problems involving differential equations. Here, we compute the Laplace transform of the polynomial function
f(t) = t² + 2t + 1, a frequent exam question in GATE, IIT JAM, CUET, and engineering mathematics. Follow the derivation, compare all options, and find the correct choice.
Step-by-Step Solution
We use standard Laplace transforms:
| Function | Laplace Transform |
|---|---|
| 1 | 1/s |
| t | 1/s² |
| t² | 2/s³ |
Using linearity:
L(t² + 2t + 1) = L(t²) + 2L(t) + L(1)
Substituting the results:
= 2/s³ + 2(1/s²) + 1/s
= 2/s³ + 2/s² + 1/s
Checking Every Option
Option (A)
1/s³ + 3/s² + 1/s — Incorrect. Coefficient of t² should be 2, not 1.
Option (B)
4/s³ + 4/s² + 1/s — Incorrect. All coefficients are doubled incorrectly.
Option (C)
1/s³ + 2/s² + 1/s — Incorrect. Coefficient of t² should be 2, not 1.
Option (D) — Correct
2/s³ + 2/s² + 1/s — Matches the derived expression exactly.
Final Answer
Correct Option: (D)
Exam Tip
Memorize:
L(tⁿ) = n! / sⁿ⁺¹
This makes Laplace problems fast and simple.
