The given expression is an integral:

\[
\int e^x \sin(e^x) \, dx
\]

To solve this, we'll use **substitution**.

### Step 1: Substitution
Let \( u = e^x \), then the derivative of \( u \) with respect to \( x \) is:

\[
du = e^x \, dx
\]

Thus, the integral becomes:

\[
\int \sin(u) \, du
\]

### Step 2: Solve the new integral
We now solve:

\[
\int \sin(u) \, du = -\cos(u) + C
\]

### Step 3: Substitute back
Since \( u = e^x \), substitute back into the result:

\[
-\cos(e^x) + C
\]

So, the solution to the integral is:

\[
\boxed{-\cos(e^x) + C}
\]

Would you like more details or have any questions? Here are some related questions:

1. How does substitution work in integration?
2. What are the general techniques to solve trigonometric integrals?
3. What is the integral of \( e^x \sin(e^x) \) when bounds are given?
4. Can integration by parts be used in this scenario?
5. What other types of functions involve exponentials and trigonometric terms?

**Tip**: Always check if substitution simplifies a complex integral before trying other methods.

Learn how to solve the integral ∫ e^x * sin(e^x) dx using substitution. This detailed solution covers key concepts such as exponential functions, trigonometric integrals, and the substitution rule for integration. Ideal for students in Grades 11-12 or introductory college-level calculus courses.

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