The given integral is:

\[
\int_{0}^{t} e^{-s} \sin(t - s) \, ds
\]

To solve this, we can use the **convolution theorem** or directly integrate by parts. Let's proceed using integration by parts:

### Step 1: Recognize the integral as a convolution
We see that this is a convolution of the functions \( e^{-s} \) and \( \sin(t - s) \).

### Step 2: Change the variable
Let \( u = t - s \), hence \( du = -ds \). The limits of integration change accordingly:
- When \( s = 0 \), \( u = t \)
- When \( s = t \), \( u = 0 \)

So, the integral becomes:
\[
\int_{t}^{0} e^{-(t-u)} \sin(u) (-du)
\]
or equivalently:
\[
\int_{0}^{t} e^{-(t-u)} \sin(u) \, du
\]

### Step 3: Factor out \( e^{-t} \)
Since \( e^{-t} \) is independent of \( u \), it can be factored out of the integral:
\[
e^{-t} \int_{0}^{t} e^{u} \sin(u) \, du
\]

### Step 4: Solve the integral
Now, we need to solve:
\[
\int e^{u} \sin(u) \, du
\]
This can be done using integration by parts, applying it twice.

Let's use the tabular method for brevity:
- Differentiate \( e^u \) until it becomes zero.
- Integrate \( \sin(u) \) repeatedly.

However, for simplicity, the result is well-known:
\[
\int e^{u} \sin(u) \, du = \frac{e^u (\sin(u) - \cos(u))}{2} + C
\]

Applying this to our problem, and considering the definite integral:
\[
\int_{0}^{t} e^{u} \sin(u) \, du = \frac{1}{2} \left[ e^{t} (\sin(t) - \cos(t)) - (0) \right]
\]

### Step 5: Substitute back and simplify
Substituting this back into the expression:
\[
e^{-t} \cdot \frac{1}{2} \left[ e^{t} (\sin(t) - \cos(t)) \right] = \frac{1}{2} \left( \sin(t) - \cos(t) \right)
\]

### Final Answer:
Thus, the solution to the integral is:

\[
\int_{0}^{t} e^{-s} \sin(t - s) \, ds = \frac{1}{2} (\sin(t) - \cos(t))
\]

Would you like more details or have any questions?

Here are some related questions:
1. How does the convolution theorem apply to solving integrals?
2. What is the tabular method in integration by parts?
3. How do you use substitution to solve definite integrals?
4. What are some common integrals involving sine and cosine functions?
5. How does the solution change if we modify the limits of integration?

**Tip:** When stuck on a complex integral, look for patterns that might simplify the problem, such as recognizing a convolution or using a known integral formula.

Learn how to solve the integral ∫0 to t e^-s sin(t - s) ds step-by-step. This problem involves integration techniques such as integration by parts and understanding the convolution theorem. Suitable for advanced undergraduate students and beyond.

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