1. The problem is to find the integral $$\int e^x \cdot \sin(e^x) \, dx$$. 2. We use substitution to solve this integral. Let $$u = e^x$$. Then, the derivative is $$\frac{du}{dx} = e^x$$, which means $$du = e^x dx$$. 3. Substitute into the integral: $$\int e^x \sin(e^x) \, dx = \int \sin(u) \, du$$. 4. The integral of $$\sin(u)$$ is $$-\cos(u) + C$$. 5. Substitute back $$u = e^x$$ to get the final answer: $$-\cos(e^x) + C$$. This completes the integration.