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We are tasked with finding the Laplace transform \( L \left\{ \int_0^t u \sinh(2u) \, du \right\} \).

### Step-by-step approach:

1. **Laplace Transform of an Integral**:
   The Laplace transform of an integral \( \int_0^t f(u) \, du \) is given by:
   \[
   L \left\{ \int_0^t f(u) \, du \right\} = \frac{1}{s} L\{f(t)\}
   \]
   where \( L\{f(t)\} \) is the Laplace transform of the function \( f(t) \).

2. **Function to Transform**:
   Here, \( f(u) = u \sinh(2u) \).

3. **Laplace Transform of \( u \sinh(2u) \)**:
   The Laplace transform of \( \sinh(2u) \) is:
   \[
   L\{\sinh(2u)\} = \frac{2}{s^2 - 4}
   \]
   Using the property \( L\{u f(u)\} = -\frac{d}{ds} L\{f(u)\} \), we find the Laplace transform of \( u \sinh(2u) \):
   \[
   L\{u \sinh(2u)\} = -\frac{d}{ds} \left( \frac{2}{s^2 - 4} \right)
   \]
   Differentiate with respect to \( s \):
   \[
   -\frac{d}{ds} \left( \frac{2}{s^2 - 4} \right) = -\frac{2 \cdot (-2s)}{(s^2 - 4)^2} = \frac{4s}{(s^2 - 4)^2}
   \]

4. **Final Step**:
   Using the formula for the Laplace transform of an integral, we divide by \( s \):
   \[
   L \left\{ \int_0^t u \sinh(2u) \, du \right\} = \frac{1}{s} \cdot \frac{4s}{(s^2 - 4)^2} = \frac{4}{(s^2 - 4)^2}
   \]

### Conclusion:
The correct answer is \( \boxed{c} \).

Let me know if you would like more details or have any questions!

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Here are 5 related questions to expand your understanding:
1. How do you compute the Laplace transform of \( \sinh(au) \) in general?
2. What is the general rule for differentiating a Laplace transform with respect to \( s \)?
3. Can you derive the Laplace transform for any other hyperbolic functions like \( \cosh(au) \)?
4. How would the result change if the integrand were \( u \cosh(2u) \) instead of \( u \sinh(2u) \)?
5. What are some practical applications of using the Laplace transform in solving differential equations?

**Tip**: When solving Laplace transforms involving products of functions, remember to use the differentiation property for terms involving \( u \).

Find the Laplace transform of the integral \( \int_0^t u \sinh(2u) \, du \).

Find the Laplace transform of the integral involving \( u \sinh(2u) \) using properties of Laplace transforms. This solution covers the Laplace differentiation theorem and integral properties. Suitable for advanced undergraduate students.

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