Test: Laplace Transform- 1 - Electronics and Communication Engineering (ECE) MCQ

We know that,

Here, f(t) = sin⁡(2t) => F(s) =
Hence,  L(t2sin(2t)) =

Key Points

• The Laplace transformation of a function f(t) is used to transform a time-domain function into a complex frequency-domain function.
• The correct definition of the Laplace transformation is given by the integral ∫₀^∞ f(t) e^-st dt. This is the correct mathematical expression for the Laplace transform.
• In this integral, f(t) is the original function of time t, and s is a complex number frequency parameter (s = σ + jω).
• The exponential term e^-st serves to weight the function f(t) and transform it into the complex frequency domain.

Additional Information

• Incorrect Options Explanation
  • ∫₀^∞ (1)/(f(t) e^st) dt: This expression is incorrect as it involves an inverse function and an incorrect exponential term.
  • ∫₀^∞ f(t) e^st dt: This is incorrect because the exponential term should be e^-st for the Laplace transformation.
  • ∫₀^∞ (1)/(f(t) e^-st) dt: This is incorrect due to the inverse function and incorrect formulation.
• Key Topics Related to Laplace Transform
  • The Laplace transform is widely used in engineering and physics to solve differential equations, control systems, and signal processing.
  • It helps in converting differential equations into algebraic equations which are easier to handle.
  • Inverse Laplace Transform: This is used to convert the function back from the frequency domain to the time domain.
  • Properties of Laplace Transform: Linearity, time shifting, frequency shifting, and convolution theorem are some key properties.
  • Applications: Commonly used in electrical engineering for circuit analysis, in mechanical engineering for system dynamics, and in control systems for stability analysis.

Given that, y(0) = 0

Now, the solution becomes,

Apply Laplace transform