Question

The Laplace transform of a unit step function \( u(t) \) is:

• A. \( \dfrac{1}{s} \)
• B. \( \dfrac{1}{s^2} \)
• C. \( s \)
• D. \( e^{-s} \)

Hint:

Always remember the basic Laplace pairs: \( u(t) \rightarrow \dfrac{1}{s} \) and \( t \rightarrow \dfrac{1}{s^2} \).

Answer 1

The Correct Option is A

Step 1: Understanding the unit step function.
The unit step function \( u(t) \) is defined as:
\[ u(t)= \begin{cases} 0, & t < 0 \\ 1, & t \geq 0 \end{cases} \] It represents a signal that switches ON at \( t = 0 \) and remains constant thereafter.
Step 2: Definition of Laplace transform.
The Laplace transform of a function \( f(t) \) is given by:
\[ \mathcal{L}\{f(t)\} = \int_{0}^{\infty} f(t)\, e^{-st}\, dt \]
Step 3: Applying Laplace transform to \( u(t) \).
Since \( u(t) = 1 \) for \( t \geq 0 \), we substitute \( f(t) = 1 \):
\[ \mathcal{L}\{u(t)\} = \int_{0}^{\infty} e^{-st}\, dt \] Evaluating the integral:
\[ \int_{0}^{\infty} e^{-st}\, dt = \left[ \frac{-1}{s} e^{-st} \right]_{0}^{\infty} = \frac{1}{s} \]
Step 4: Conclusion.
The Laplace transform of the unit step function is \( \dfrac{1}{s} \).