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.