1. The problem is to evaluate the integral $$\int e^x \sin(x) \, dx$$. 2. We use the method of integration by parts or recognize this as a standard integral involving the product of an exponential and a trigonometric function. 3. Recall the formula for integrals of the form $$\int e^{ax} \sin(bx) \, dx$$ or $$\int e^{ax} \cos(bx) \, dx$$: $$\int e^{ax} \sin(bx) \, dx = \frac{e^{ax}}{a^2 + b^2} (a \sin(bx) - b \cos(bx)) + C$$ 4. In our case, $$a = 1$$ and $$b = 1$$, so: $$\int e^x \sin(x) \, dx = \frac{e^x}{1^2 + 1^2} (1 \cdot \sin(x) - 1 \cdot \cos(x)) + C = \frac{e^x}{2} (\sin(x) - \cos(x)) + C$$ 5. Therefore, the final answer is: $$\boxed{\int e^x \sin(x) \, dx = \frac{e^x}{2} (\sin(x) - \cos(x)) + C}$$