It looks like you're working with an integral expression that might involve some typographical or formatting issues. Let me break it down as best as I can based on the symbols provided.

The integral expression seems to be: $I = \int_{0}^{4} (te^{-t}) \, dt$ This is an integral of the function $te^{-t}$ from $t = 0$ to $t = 4$.

To solve this, you can use integration by parts.

Step-by-Step Solution:

1. Set up the integration by parts formula: Recall the integration by parts formula: $\int u \, dv = uv - \int v \, du$

2. Choose $u$ and $dv$: For $\int te^{-t} \, dt$, let:

   • $u = t$ (so that $du = dt$)
   • $dv = e^{-t} dt$ (so that $v = -e^{-t}$)

3. Apply the formula: Now, applying the formula: $\int te^{-t} \, dt = -t e^{-t} - \int (-e^{-t}) \, dt$ Simplifying: $\int te^{-t} \, dt = -t e^{-t} + \int e^{-t} \, dt$

4. Integrate the remaining part: The integral of $e^{-t}$ is $-e^{-t}$, so: $\int te^{-t} \, dt = -t e^{-t} - e^{-t}$

5. Evaluate the definite integral: Now evaluate from $t = 0$ to $t = 4$: $I = \left[ -t e^{-t} - e^{-t} \right]_{0}^{4}$

6. Substitute the limits: First, at $t = 4$: $-4 e^{-4} - e^{-4} = -5e^{-4}$ And at $t = 0$: $-0 \cdot e^{0} - e^{0} = -1$

7. Final result: $I = (-5e^{-4}) - (-1) = -5e^{-4} + 1$

So, the value of the integral is: $I = 1 - 5e^{-4}$

If you want me to go over any specific part in more detail, feel free to ask!

Related Questions:

1. How do you apply integration by parts for other functions?
2. What are the properties of exponential functions in integration?
3. How do you handle limits of integration that go to infinity?
4. Can integration by parts be used for trigonometric integrals?
5. How do you estimate integrals numerically?

Tip:

When using integration by parts, always try to pick $u$ as the function that simplifies when differentiated, and $dv$ as the part that integrates easily.