The integral you're interested in is:

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

### Step 1: Simplify the Integral
The integral can be expressed as:

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

### Step 2: Substitute \( u = t - s \)
To simplify the integration, let’s make the substitution:

\[
u = t - s \quad \text{so that} \quad du = -ds
\]

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

Substituting these into the integral:

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

This simplifies to:

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

### Step 3: Interpret the Integral
The integral:

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

does not have an elementary antiderivative, which means it cannot be expressed in terms of basic functions like polynomials, exponentials, or trigonometric functions. Typically, this integral would be evaluated numerically or left in its integral form for specific values of \( t \).

### Final Answer
The integral can be expressed as:

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

Or equivalently:

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

This is the most simplified form.

Would you like to see how this integral could be evaluated numerically or discuss its applications?

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Here are 5 related questions that might be interesting:

1. How would you evaluate this integral numerically for a specific value of \( t \)?
2. What is the significance of integrals involving trigonometric functions in exponential terms?
3. Can integrals of this form be approximated using series expansion methods?
4. What role does the substitution \( u = t - s \) play in simplifying integrals?
5. Are there other mathematical techniques for dealing with non-elementary integrals?

**Tip:** For integrals that do not have simple antiderivatives, considering numerical methods or approximate techniques like series expansions can be very helpful.

Explore the integral \( \int_0^t -e^{8\sin(t-s)} \, ds \) and its simplification using substitution and properties of exponential and trigonometric functions. Learn its significance and applications in mathematical analysis.

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