Laplace Transform Calculation
The Laplace transform of a function $f(t)$ is defined as:
$F(s) = \mathcal{L}[f(t)] = \int_{0}^{\infty} e^{-st} f(t) dt$
Given the function $f(t) = e^{-at}$, we substitute it into the definition:
$F(s) = \int_{0}^{\infty} e^{-st} (e^{-at}) dt$
Evaluating the Integral
Combine the exponents:
$F(s) = \int_{0}^{\infty} e^{-(s+a)t} dt$
Evaluate the definite integral. This requires $s+a > 0$ for convergence:
$F(s) = \left[ \frac{e^{-(s+a)t}}{-(s+a)} \right]_{0}^{\infty}$
Apply the limits of integration:
$F(s) = \lim_{t\to\infty} \left( \frac{e^{-(s+a)t}}{-(s+a)} \right) - \left( \frac{e^{0}}{-(s+a)} \right)$
Since $s+a > 0$, the limit term approaches 0:
$F(s) = 0 - \left( \frac{1}{-(s+a)} \right)$
$F(s) = \frac{1}{(s+a)}$
Conclusion
The calculated Laplace transform $F(s)$ for the function $f(t) = e^{-at}$ is $\frac{1}{(s+a)}$.
This matches Option 4 (Option D) provided in the question.
Important Questions from Laplace Transform
Which of the following is the final value of the impulse response of the system whose transfer function is
(2s + 1)/(s 4 + 8s 3 + 16s 2 + s)
Find the Laplace transform for the following time domain.
y(t) = -2te -t + 4e -t - 4e -2t
Match List I with List II
List – I
List – II
f(t)
F(S)
A.
e -at
I.
\(\rm \frac{s}{s^2+ \omega^2}\)
B.
te at
II.
\(\rm \frac{\omega}{s^2+ \omega^2}\)
C.
sinωt
III.
\(\rm \frac{1}{(s- a)^2}\)
D.
cosωt
IV.
\(\rm \frac{1}{(s+ a)}\)
Choose the correct answer from the options given below:
The Laplace transform of sin h (at) is
The unilateral Laplace transform of f(t) is \(\frac{1}{s^2+s+1}\). The unilateral Laplace transform of t f(t) is
