To evaluate the integral

$\int_{0}^{\infty} e^{-2t} \left( \int_{0}^{t} \frac{e^{3u} \sin^2(2u)}{u} \, du \right) \, dt$

we can use the Laplace transform technique. Let's break down the process step-by-step.

Step 1: Define the Inner Integral

Let $f(t) = \int_{0}^{t} \frac{e^{3u} \sin^2(2u)}{u} \, du$ Then our integral becomes $\int_{0}^{\infty} e^{-2t} f(t) \, dt$

Step 2: Laplace Transform of the Outer Integral

The Laplace transform of $e^{-2t} f(t)$ is given by: $\mathcal{L}\{ e^{-2t} f(t) \}(s) = \mathcal{L}\{ f(t) \}(s + 2)$ Thus, we need to compute the Laplace transform $\mathcal{L}\{ f(t) \}(s)$ and then evaluate it at $s + 2$.

Step 3: Laplace Transform of $f(t)$

The Laplace transform of $f(t) = \int_{0}^{t} g(u) \, du$ is: $\mathcal{L}\{ f(t) \}(s) = \frac{\mathcal{L}\{ g(t) \}(s)}{s}$ where $g(t) = \frac{e^{3t} \sin^2(2t)}{t}$.

Step 4: Find $\mathcal{L}\{ g(t) \}(s)$

To proceed further, we need the Laplace transform of $g(t) = \frac{e^{3t} \sin^2(2t)}{t}$. This is not a straightforward transform, and direct computation might involve advanced techniques or tables of Laplace transforms for such functions.

Given the complexity, we typically refer to known results or tables for Laplace transforms of functions of this type, as exact calculation could be challenging without specialized techniques.

Final Answer

Because of the complexity in finding $\mathcal{L}\{ g(t) \}(s)$, this integral is generally approached with advanced techniques or numerical methods rather than by hand. The Laplace transform approach is a suitable framework, but evaluating it fully by hand would require more detailed information on $\mathcal{L}\{ g(t) \}(s)$.

If you need further assistance with any part of this problem, feel free to ask. Here are some questions to deepen your understanding:

1. Would you like an explanation of Laplace transforms and their properties?
2. Do you want to explore alternative methods for evaluating this integral?
3. Are you interested in a step-by-step explanation of the Laplace transform process?
4. Would you like a summary of common Laplace transform pairs?
5. Do you need assistance with the inner integral's properties or simplifications?

Tip: Laplace transforms are particularly useful for handling integrals involving exponentials and step functions.