1. **Problem:** Find the integral $$I_1 = \int 2x e^x \, dx$$. 2. **Formula and method:** Use integration by parts, which states: $$\int u \, dv = uv - \int v \, du$$ Choose: $$u = 2x \implies du = 2 \, dx$$ $$dv = e^x \, dx \implies v = e^x$$ 3. **Apply integration by parts:** $$I_1 = 2x e^x - \int e^x \cdot 2 \, dx = 2x e^x - 2 \int e^x \, dx$$ 4. **Evaluate the remaining integral:** $$\int e^x \, dx = e^x + C$$ So, $$I_1 = 2x e^x - 2 e^x + C$$ 5. **Final answer:** $$\boxed{I_1 = e^x (2x - 2) + C}$$ This completes the solution for the first integral.