It seems you're dealing with a vector-valued integral, where the integral is taken over a vector function. Let's break it down carefully:

You have the vector function:

$\mathbf{F}(t) = 8t \mathbf{i} - t^3 \mathbf{j} + 4t^7 \mathbf{k}$

And the integral is expressed as:

$\int_0^2 \mathbf{F}(t) \, dt$

This means you want to integrate the components of the vector function with respect to $t$ from $0$ to $2$.

Step 1: Split into components

We will handle the components of the vector function individually.

$\mathbf{F}(t) = \langle 8t, -t^3, 4t^7 \rangle$

So, the integral becomes:

$\int_0^2 \langle 8t, -t^3, 4t^7 \rangle dt = \left\langle \int_0^2 8t \, dt, \int_0^2 -t^3 \, dt, \int_0^2 4t^7 \, dt \right\rangle$

Step 2: Evaluate the individual integrals

1. For the first component $8t$: $\int_0^2 8t \, dt = 8 \left[ \frac{t^2}{2} \right]_0^2 = 8 \times \frac{4}{2} = 16$

2. For the second component $-t^3$: $\int_0^2 -t^3 \, dt = - \left[ \frac{t^4}{4} \right]_0^2 = - \frac{16}{4} = -4$

3. For the third component $4t^7$: $\int_0^2 4t^7 \, dt = 4 \left[ \frac{t^8}{8} \right]_0^2 = 4 \times \frac{256}{8} = 128$

Step 3: Combine the results

Now, combining all the components, we get the result of the vector integral:

$\int_0^2 \mathbf{F}(t) \, dt = \langle 16, -4, 128 \rangle$

Final Answer:

$\int_0^2 \mathbf{F}(t) \, dt = \langle 16, -4, 128 \rangle$

Let me know if you need further clarification or any additional questions!

---

Here are some related questions you might find useful:

1. What happens if the vector function changes form (e.g., a different exponent or coefficient)?
2. How do we handle integrals of vector functions with more than one variable (multivariable)?
3. What is the physical meaning of such integrals in a real-world context (e.g., work or force)?
4. How can we handle integrals over more complex paths or curves?
5. How do we interpret results of a vector-valued integral in terms of direction and magnitude?

Tip: Always ensure the components of the vector are handled individually and check the limits of integration to avoid mistakes!