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To solve this problem, we need to find the Laplace transform of the integral:

\[
L\left\{ \int_0^t u \sin 2u \, du \right\}
\]

### Solution Steps

1. **Recognize the Laplace Transform Property**:
   For an integral of the form \( F(t) = \int_0^t f(u) \, du \), the Laplace transform \( L\{F(t)\} = \frac{L\{f(t)\}}{s} \).

2. **Set up the function \( f(u) = u \sin 2u \)**:
   We need the Laplace transform of \( u \sin 2u \), which we denote as \( L\{u \sin 2u\} \).

3. **Use the Laplace Transform of \( t \sin(at) \)**:
   The Laplace transform \( L\{t \sin(at)\} = \frac{2a s}{(s^2 + a^2)^2} \).

   Here, \( a = 2 \), so the transform of \( u \sin 2u \) is:
   \[
   L\{u \sin 2u\} = \frac{4s}{(s^2 + 4)^2}
   \]

4. **Apply the Integral Property**:
   Since \( F(t) = \int_0^t u \sin 2u \, du \), we divide by \( s \):
   \[
   L\left\{ \int_0^t u \sin 2u \, du \right\} = \frac{\frac{4s}{(s^2 + 4)^2}}{s} = \frac{4s}{s(s^2 + 4)^2}
   \]

5. **Simplify**:
   This simplifies to:
   \[
   \frac{4}{(s^2 + 4)^2}
   \]

### Conclusion

The correct answer is **(c) \(\frac{4}{(s^2 + 4)^2}\)**.

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Do you want more details or have any questions?

Here are some related questions for further practice:

1. How would you find the Laplace transform of \( \int_0^t t^2 \cos(3t) \, dt \)?
2. What is the Laplace transform of \( t^2 \sin(5t) \)?
3. How can we use Laplace transforms to solve differential equations with integrals?
4. What is the Laplace transform of \( \int_0^t e^{3u} \cos(u) \, du \)?
5. How do you derive the Laplace transform formula for \( t \sin(at) \)?

**Tip:** Remember that the Laplace transform of an integral can be simplified by dividing by \( s \) after transforming the inner function.

Find the Laplace transform of \( \int_0^t u \sin 2u \, du \)

Step-by-step solution to finding the Laplace transform of \( \int_0^t u \sin 2u \, du \). This problem covers integral properties of Laplace transforms, the Laplace transform of \( t \sin(at) \), and applies advanced calculus concepts. Suitable for university-level calculus students.

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