To integrate \( \sin(x) e^{\cos(x)} \), we can use the method of substitution. Here's the step-by-step process:

### Step 1: Substitution
Let:
\[
u = \cos(x)
\]
Then, the derivative of \( u \) with respect to \( x \) is:
\[
\frac{du}{dx} = -\sin(x)
\]
This gives us:
\[
du = -\sin(x) \, dx
\]
or:
\[
-\sin(x) \, dx = du
\]

### Step 2: Rewrite the integral
Using this substitution, we rewrite the integral \( \int \sin(x) e^{\cos(x)} \, dx \) as:
\[
\int e^{u} (-du)
\]
This simplifies to:
\[
-\int e^{u} \, du
\]

### Step 3: Integrate
The integral of \( e^u \) with respect to \( u \) is simply \( e^u \). Therefore:
\[
-\int e^u \, du = -e^u + C
\]
where \( C \) is the constant of integration.

### Step 4: Substitute back
Now, substitute \( u = \cos(x) \) back into the result:
\[
-e^{\cos(x)} + C
\]

### Final Answer:
\[
\int \sin(x) e^{\cos(x)} \, dx = -e^{\cos(x)} + C
\]

Would you like any additional details or clarifications?

Here are 5 related questions to explore this concept further:
1. How do you determine the appropriate substitution in an integral?
2. Can this technique be applied to other trigonometric integrals involving exponentials?
3. How would you solve \( \int \cos(x) e^{\sin(x)} \, dx \)?
4. How do you handle integrals involving products of different functions?
5. What are the common strategies for integrating functions involving both sine and cosine?

**Tip:** Always check if substitution is a viable method by looking for a function and its derivative in the integral.

Learn how to solve the integral of sin(x) e^cos(x) with a step-by-step substitution method. This calculus problem explores the use of substitution to simplify trigonometric and exponential integrals. Suitable for undergraduate calculus students.

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