6.2. Theorem: Laplace Transform of an Integral

Try these first

1. 1.

   What is the basic definition of the Laplace Transform?

   Hint

   Recall the integral form involving an exponential function.

2. 2.

   If L{f(t)} = F(s), what is the result of L{∫₀^t f(τ)dτ}?

   Hint

   Think about how integration translates in the Laplace domain.

3. 3.

   What operation corresponds to integrating in the Laplace domain?

   • Multiply by s
   • Divide by s
   • Subtract s

   Hint

   Think about how integration alters transformation.

4. 4.

   True or False: The Laplace Transform can simplify solving integro-differential equations.

   • True
   • False

   Hint

   Recall the applications of this theorem.

5. 5.

   Using the Laplace Transform, solve for the integral of x(t) = e^{-3t} in the interval [0, t].

   Hint

   Start with the known transform of e^{-3t}.

6. 6.

   If L{f(t)} results in a rational function with a singularity, how would it affect L{∫₀^t f(τ)dτ} due to its division by a variable factor?

   Hint

   Think of the nature of discontinuities and their results in the integration context.