1. **State the problem:** We want to evaluate the integral $$\int y e^y \, dy$$. 2. **Choose a method:** Use integration by parts, where we let: - $$u = y$$ so that $$du = dy$$ - $$dv = e^y dy$$ so that $$v = e^y$$ 3. **Apply integration by parts formula:** $$\int u \, dv = uv - \int v \, du$$ Substitute the chosen parts: $$\int y e^y \, dy = y e^y - \int e^y \, dy$$ 4. **Integrate the remaining integral:** $$\int e^y \, dy = e^y + C$$ 5. **Combine results:** $$\int y e^y \, dy = y e^y - e^y + C = e^y (y - 1) + C$$ 6. **Final answer:** $$\boxed{\int y e^y \, dy = e^y (y - 1) + C}$$