1. Problem: Evaluate the integral $\int x e^x \, dx$ using integration by parts with $u = x$ and $dv = e^x dx$. 2. Formula: Integration by parts states: $$\int u \, dv = uv - \int v \, du$$ 3. Step 1: Identify $u = x$ and $dv = e^x dx$. 4. Step 2: Compute derivatives and integrals: $$du = dx$$ $$v = \int e^x dx = e^x$$ 5. Step 3: Apply the formula: $$\int x e^x dx = x e^x - \int e^x dx$$ 6. Step 4: Evaluate the remaining integral: $$\int e^x dx = e^x$$ 7. Step 5: Substitute back: $$\int x e^x dx = x e^x - e^x + C$$ 8. Final answer: $$\boxed{e^x (x - 1) + C}$$ This completes the solution for the first problem.