. Evaluate ∫ 𝑡
∞
0
𝑒
–3t sin 2𝑡 𝑑�

To evaluate the integral 

\[
\int_0^{\infty} e^{-3t} \sin(2t) \, dt,
\]

we can use the formula for the Laplace transform of \( e^{at} \sin(bt) \). The general formula for the integral 

\[
\int_0^{\infty} e^{-pt} \sin(qt) \, dt
\]

is given by:

\[
\frac{q}{p^2 + q^2}
\]

where \( p > 0 \).

In this case:

- \( p = 3 \)
- \( q = 2 \)

Substitute these values into the formula:

\[
\int_0^{\infty} e^{-3t} \sin(2t) \, dt = \frac{2}{3^2 + 2^2} = \frac{2}{9 + 4} = \frac{2}{13}
\]

So, the value of the integral is:

\[
\frac{2}{13}
\]

Would you like a detailed step-by-step explanation or have any questions?

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Here are 5 related questions to practice:

1. Evaluate \(\int_0^{\infty} e^{-2t} \cos(3t) \, dt\).
2. Find the Laplace transform of \( e^{-3t} \cos(4t) \).
3. Determine the integral \(\int_0^{\infty} e^{-t} \sin(5t) \, dt\).
4. Calculate \(\int_0^{\infty} e^{-4t} \sin(2t) \, dt\).
5. What is the value of \(\int_0^{\infty} e^{-2t} \sin(t) \, dt\)?

**Tip:** Understanding the Laplace transform formulas can greatly simplify the evaluation of many improper integrals involving exponentials and trigonometric functions.

This page provides a detailed solution for evaluating the integral ∫₀^∞ e^(-3t) sin(2t) dt using the Laplace transform. Learn how to apply the formula for improper integrals involving exponential decay and trigonometric functions, suitable for students in Calculus II or advanced calculus.

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