To solve this problem, we need to determine the Laplace transform of the integral $\int_0^t u \sin(2u) \, du$, given by:

$L \left\{ \int_0^t u \sin(2u) \, du \right\}.$

Steps to Solution

1. Using the Laplace Transform of an Integral: For an integral of the form $\int_0^t f(u) \, du$, the Laplace transform is given by: $L \left\{ \int_0^t f(u) \, du \right\} = \frac{F(s)}{s},$ where $F(s) = L\{f(u)\}$.

2. Determine $f(u)$ and Its Laplace Transform: Here, $f(u) = u \sin(2u)$.

   To find $L\{u \sin(2u)\}$, we use the Laplace transform property for $t \sin(at)$, which is: $L\{t \sin(at)\} = \frac{2a}{(s^2 + a^2)^2}.$ In our case, $a = 2$, so: $L\{u \sin(2u)\} = \frac{4}{(s^2 + 4)^2}.$

3. Apply the Integral Formula: Now that we have $F(s) = \frac{4}{(s^2 + 4)^2}$, applying the formula gives: $L \left\{ \int_0^t u \sin(2u) \, du \right\} = \frac{4}{s(s^2 + 4)^2}.$

Finding the Correct Answer

The correct answer matches the expression we derived:

$\frac{4}{s(s^2 + 4)^2}.$

After checking the options, the correct choice is option (c).

Would you like further explanation or have additional questions?

Here are some related questions:

1. How do you find the Laplace transform of other integrals involving $t$ and trigonometric functions?
2. What are the steps to apply the convolution theorem in Laplace transforms?
3. How can the Laplace transform be used in solving differential equations?
4. What is the significance of the Laplace transform in control systems?
5. How do you handle Laplace transforms of higher-order polynomial functions of $t$?

Tip: When finding the Laplace transform of integrals, focus on the properties and theorems that simplify the process, such as the integral formula and convolution theorem.