Laplace Transform of f(t) = (cos(3t) - 1) / t | | Integration Property (Division by t)

#laplacetransform #laplacetransformation #maths #manim #education #visualization #calculus #differentialequations #mathematics #math #engineeringmath #engineeringmathematics #laplace #transform
#integration #property #division #engineering #signal #processing #solved
#examples #logarithmicfunction #functions #improperintegrals #improper


In this video, we explore the Integration Property (Division by t) of the Laplace Transform.

We start by stating the theorem:
L{f(t)/t} = ∫ F(σ) dσ from p to ∞.

Then, we solve a specific example step-by-step:
Find the Laplace transform of f(t) = (cos(3t) - 1) / t.

Timeline:
0:00 - Theorem: Integration Property of Laplace Transform
0:35 - Example Setup: L{(cos(3t)-1)/t}
2:45 - Visualization: Time Domain f(t) vs s-Domain F(p)

The final result is shown to be F(p) = ln(p / sqrt(p² + 9)).
We conclude with a graphical comparison showing how the damped oscillation in the time domain corresponds to the logarithmic decay in the s-domain.

   • Theorem: Every Convergent Sequence is Boun...  

   • Solving Differential Equation xy\,dx + (x+...  

   • Does it Converge or Diverge? Integral of 1...  

   • Integral of 1/x^2: Power Rule with Negativ...  

   • Complex Analysis: Finding the Laurent Seri...  

   • Metric Spaces: Definition, Axioms, and Exa...