15.7.4. Integration Theorem

Try these first

1. 1.

   What does the Integration Theorem state about the Laplace transform of an integral?

   Hint

   Think about how integration relates to transformation.

2. 2.

   In which cases is the Integration Theorem particularly useful?

   Hint

   Consider the context of engineering and applications.

3. 3.

   What is the result of applying the Integration Theorem to a function f(t) with its Laplace transform?

   • F(s)
   • F(s)/s
   • sF(s)

   Hint

   Remember how integration modifies the transformation.

4. 4.

   True or False: The Integration Theorem cannot be applied to functions that are not continuous.

   • True
   • False

   Hint

   Think about the conditions under which the theorem is valid.

5. 5.

   Given the differential equation y'' + 6y' + 9y = 0 with initial conditions y(0)=1 and y'(0)=0, apply the Integration Theorem to solve for y(t).

   Hint

   Break the problem down into the Laplace steps and think about initial values.

6. 6.

   Using the function f(t) = t^2, apply the Integration Theorem to find its Laplace transform.

   Hint

   Start with the integration and then transform!