1. Problem A: Integrate $\int x e^{2x} \, dx$ using integration by parts with $u = x$ and $dv = e^{2x} \, dx$. 2. Recall the integration by parts formula: $$\int u \, dv = uv - \int v \, du$$ 3. Compute $du$ and $v$: - $du = dx$ - To find $v$, integrate $dv$: $v = \int e^{2x} \, dx = \frac{1}{2} e^{2x}$ 4. Apply the formula: $$\int x e^{2x} \, dx = x \cdot \frac{1}{2} e^{2x} - \int \frac{1}{2} e^{2x} \, dx$$ 5. Simplify the remaining integral: $$= \frac{x}{2} e^{2x} - \frac{1}{2} \int e^{2x} \, dx = \frac{x}{2} e^{2x} - \frac{1}{2} \cdot \frac{1}{2} e^{2x} + C = \frac{x}{2} e^{2x} - \frac{1}{4} e^{2x} + C$$ 6. Final answer: $$\int x e^{2x} \, dx = \frac{e^{2x}}{4} (2x - 1) + C$$