1. Let's start by stating the problem: Evaluate the integral $$\int e^x \sin(e^x) \, dx$$. 2. To solve this integral, we use substitution. Let $$u = e^x$$. Then, the derivative is $$\frac{du}{dx} = e^x$$, which means $$du = e^x \, dx$$. 3. Notice that in the integral, we have $$e^x \, dx$$, which matches exactly with $$du$$. So, the integral becomes: $$\int \sin(u) \, du$$ 4. The integral of $$\sin(u)$$ with respect to $$u$$ is: $$-\cos(u) + C$$ 5. Now, substitute back $$u = e^x$$ to get the final answer: $$-\cos(e^x) + C$$ 6. To address your question about "losing" one $$e^x$$: When we substitute, the $$e^x \, dx$$ part is replaced by $$du$$. This means the $$e^x$$ inside the integral is accounted for in the substitution and does not disappear; it is transformed into the differential $$du$$. So, the other $$e^x$$ is not lost but incorporated into the substitution process. Therefore, the integral evaluates to: $$-\cos(e^x) + C$$