The integral you're trying to solve is:

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

Here, \( t \) is treated as a constant with respect to \( s \), and we are integrating with respect to \( s \) over the interval from \( 0 \) to \( t \).

### Step 1: Simplify the Integral

The integral can be rewritten as:
\[
-e^8 \int_0^t \sin(t - s) \, ds
\]

### Step 2: Use Substitution

We can use the substitution \( u = t - s \). Thus:
\[
du = -ds
\]

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

The integral bounds will change accordingly:
\[
-e^8 \int_t^0 \sin(u) (-du)
\]

This simplifies to:
\[
e^8 \int_0^t \sin(u) \, du
\]

### Step 3: Integrate

The integral of \( \sin(u) \) with respect to \( u \) is:
\[
\int \sin(u) \, du = -\cos(u)
\]

So:
\[
e^8 \int_0^t \sin(u) \, du = e^8 \left[-\cos(u)\right]_0^t
\]

### Step 4: Evaluate the Definite Integral

Now, substitute the limits:
\[
e^8 \left[-\cos(t) + \cos(0)\right] = e^8 \left[-\cos(t) + 1\right]
\]

So, the final answer is:
\[
\boxed{\int_0^t -e^8 \sin(t - s) \, ds = e^8 \left(1 - \cos(t)\right)}
\]

Would you like more details, or do you have any questions?

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Here are 5 related questions you might find interesting:

1. How would you solve \(\int_0^t e^8 \cos(t - s) \, ds\)?
2. What happens if you change the bounds to \(\int_0^{\pi} -e^8 \sin(t - s) \, ds\)?
3. How do you handle integrals involving products of exponential and trigonometric functions?
4. Can you solve \(\int_0^t e^{at} \sin(bt) \, dt\) where \(a\) and \(b\) are constants?
5. How does the substitution method compare to other methods in solving definite integrals?

**Tip:** When dealing with definite integrals, make sure to adjust the bounds properly after performing substitution. This helps in avoiding mistakes in the final evaluation.

Explore the integral of -e^8 sin(t - s) from 0 to t with a detailed step-by-step solution. Learn the substitution method, trigonometric identities, and exponential functions used in solving this definite integral. Suitable for advanced college-level students and professionals.

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