To evaluate the integral

$I = \int_0^\infty e^{-\sqrt{3}t} t \sin(2t) \, \mathrm{d}t,$

we use the method of integration by parts and properties of the Laplace transform. Let's proceed step by step:

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Step 1: Recognizing the structure

The integral has the form:

$I = \int_0^\infty e^{-\alpha t} t \sin(\beta t) \, \mathrm{d}t,$

where $\alpha = \sqrt{3}$ and $\beta = 2$. This is a standard form, and we can use known results or derive it using integration by parts.

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Step 2: Using the Laplace transform

The Laplace transform of $t \sin(\beta t)$ is:

$\mathcal{L}\{t \sin(\beta t)\}(s) = \frac{2\beta s}{(s^2 + \beta^2)^2}.$

Substitute $s = \sqrt{3}$ and $\beta = 2$:

$\mathcal{L}\{t \sin(2t)\}(\sqrt{3}) = \frac{2 \cdot 2 \cdot \sqrt{3}}{(\sqrt{3}^2 + 2^2)^2}.$

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Step 3: Simplify the denominator

$\sqrt{3}^2 + 2^2 = 3 + 4 = 7,$ so: $\mathcal{L}\{t \sin(2t)\}(\sqrt{3}) = \frac{4 \sqrt{3}}{7^2}.$

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Step 4: Final result

$I = \frac{4 \sqrt{3}}{49}.$

Thus, the value of the integral is:

$\boxed{\frac{4 \sqrt{3}}{49}}.$

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Do you want further explanation or derivation for any step?

Related Questions:

1. What is the Laplace transform, and how is it derived for $t \sin(\beta t)$?
2. Can you solve this integral using direct integration by parts instead of Laplace transform?
3. What are some applications of similar integrals in engineering or physics?
4. How does the exponential term $e^{-\sqrt{3}t}$ affect the convergence of the integral?
5. How can you extend this integral to include cosine instead of sine?

Tip:

For integrals involving products of $e^{-\alpha t}$, polynomials, and trigonometric functions, the Laplace transform is often a highly efficient tool!