We are asked to evaluate the following definite integral:

$\int_0^2 \left(-3e^{-4t} + 2\right) \, dt$

Step 1: Break the Integral Into Two Parts

We can separate the given integral into two parts:

$\int_0^2 \left(-3e^{-4t} + 2\right) \, dt = \int_0^2 -3e^{-4t} \, dt + \int_0^2 2 \, dt$

Step 2: Solve the First Integral

For the first part, we have:

$\int_0^2 -3e^{-4t} \, dt$

The integral of $e^{-4t}$ is:

$\int e^{-4t} \, dt = \frac{e^{-4t}}{-4}$

Thus,

[ \int_0^2 -3e^{-4t} , dt = -3 \left[ \frac{e^{-4t}}{-4} \right]_0^2 = \frac{3}{4} \left[ e^{-4t} \right]_0^2 ]

Now, evaluate the expression:

$\frac{3}{4} \left( e^{-8} - e^0 \right) = \frac{3}{4} \left( e^{-8} - 1 \right)$

Step 3: Solve the Second Integral

For the second part, we have:

$\int_0^2 2 \, dt = 2t \Big|_0^2 = 2(2) - 2(0) = 4$

Step 4: Combine the Results

Now, combine the results of both integrals:

$\int_0^2 \left(-3e^{-4t} + 2\right) \, dt = \frac{3}{4} \left( e^{-8} - 1 \right) + 4$

Final Answer:

The value of the definite integral is:

$\boxed{\frac{3}{4} \left( e^{-8} - 1 \right) + 4}$

Would you like further explanation on any step, or do you have any questions about the result?

5 Related Questions:

1. How do you evaluate integrals involving exponential functions?
2. What is the significance of the negative sign in the first integral?
3. How do you handle constants in definite integrals?
4. What are the properties of exponential decay functions?
5. How do you approximate the value of $e^{-8}$ for large powers?

Tip: When dealing with definite integrals, always break them into simpler parts, especially if the integrand is a sum of multiple terms. This allows you to apply standard integration rules to each term individually.