The integral you're asking about is:

\[
\int_0^t -e^{8\sin(t-s)} \, ds
\]

To solve this integral, we'll treat it as a function of \( t \), where \( t \) is treated as a constant with respect to the variable of integration \( s \).

### Step 1: Rewrite the integral
The integral can be rewritten as:

\[
\int_0^t -e^{8\sin(t-s)} \, ds
\]

### Step 2: Substitute \( u = t - s \)
Let's make a substitution to simplify the integral:

Let \( u = t - s \). Then \( du = -ds \).

When \( s = 0 \), \( u = t \).  
When \( s = t \), \( u = 0 \).

So the integral becomes:

\[
\int_t^0 e^{8\sin(u)} (-du) = \int_0^t e^{8\sin(u)} \, du
\]

### Step 3: Solve the integral
The integral is now:

\[
\int_0^t e^{8\sin(u)} \, du
\]

This integral doesn't have a simple closed-form solution in elementary functions. It's usually expressed in terms of the **exponential integral** or solved numerically.

### Final Answer
Thus, the integral in its exact form is:

\[
\int_0^t -e^{8\sin(t-s)} \, ds = -\int_0^t e^{8\sin(u)} \, du
\]

Or equivalently, after changing the sign:

\[
-\int_0^t e^{8\sin(t-s)} \, ds = \int_0^t e^{8\sin(u)} \, du
\]

This is the most simplified form unless you evaluate it numerically.

Would you like to go into more details on solving this integral numerically or its properties?

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Here are 5 related questions that you might find helpful:

1. What is the exponential integral function, and how is it related to this problem?
2. How can this integral be approximated using numerical methods?
3. What are some other examples of integrals that do not have elementary closed-form solutions?
4. How does the substitution method simplify complex integrals like this one?
5. What are some practical applications of integrals involving trigonometric exponentials?

**Tip:** When an integral doesn't have a straightforward antiderivative, numerical methods or special functions like the exponential integral often come in handy.

Explore how to solve the integral ∫0^t -e^8sin(t-s) ds using substitution methods and integral properties. This problem involves integral calculus, exponential functions, and trigonometric functions.

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