1. The problem is to find the integral $$\int e^x \sin x \, dx$$. 2. We use integration by parts or recognize this as a standard integral involving exponential and trigonometric functions. 3. The formula for integrating $$e^{ax} \sin(bx)$$ is $$\int e^{ax} \sin(bx) \, dx = \frac{e^{ax}}{a^2 + b^2} (a \sin(bx) - b \cos(bx)) + C$$. 4. Here, $$a = 1$$ and $$b = 1$$, so the integral becomes: $$\int e^x \sin x \, dx = \frac{e^x}{1^2 + 1^2} (1 \cdot \sin x - 1 \cdot \cos x) + C = \frac{e^x}{2} (\sin x - \cos x) + C$$. 5. This is the final answer.