`(i) int x^(2) " cos x dx " " "(ii) int x^(2) e^(3x) " dx "`

Answer

Let's solve the given integrals step by step.

### Part (i): \(\int x^2 \cos x \, dx\)

1. **Apply Integration by Parts**:
We use the integration by parts formula:
\[ \int u \, dv = uv - \int v \, du \]
Let:
- \(u = x^2\) (thus \(du = 2x \, dx\))
- \(dv = \cos x \, dx\) (thus \(v = \sin x\))

2. **Substituting into the Formula**:
\[ \int x^2 \cos x \, dx = x^2 \sin x - \int \sin x \cdot 2x \, dx \]

3. **Simplify the Integral**:
\[ = x^2 \sin x - 2 \int x \sin x \, dx \]

4. **Apply Integration by Parts Again**:
For \(\int x \sin x \, dx\), let:
- \(u = x\) (thus \(du = dx\))
- \(dv = \sin x \, dx\) (thus \(v = -\cos x\))

5. **Substituting into the Formula**:
\[ \int x \sin x \, dx = -x \cos x - \int -\cos x \, dx = -x \cos x + \sin x \]

6. **Substituting Back**:
\[ \int x^2 \cos x \, dx = x^2 \sin x - 2(-x \cos x + \sin x) \]
\[ = x^2 \sin x + 2x \cos x - 2\sin x + C \]

### Final Answer for Part (i):
\[ \int x^2 \cos x \, dx = x^2 \sin x + 2x \cos x - 2\sin x + C \]

---

### Part (ii): \(\int x^2 e^{3x} \, dx\)

1. **Apply Integration by Parts**:
Again, we use the integration by parts formula:
Let:
- \(u = x^2\) (thus \(du = 2x \, dx\))
- \(dv = e^{3x} \, dx\) (thus \(v = \frac{1}{3} e^{3x}\))

2. **Substituting into the Formula**:
\[ \int x^2 e^{3x} \, dx = x^2 \cdot \frac{1}{3} e^{3x} - \int \frac{1}{3} e^{3x} \cdot 2x \, dx \]
\[ = \frac{1}{3} x^2 e^{3x} - \frac{2}{3} \int x e^{3x} \, dx \]

3. **Apply Integration by Parts Again**:
For \(\int x e^{3x} \, dx\), let:
- \(u = x\) (thus \(du = dx\))
- \(dv = e^{3x} \, dx\) (thus \(v = \frac{1}{3} e^{3x}\))

4. **Substituting into the Formula**:
\[ \int x e^{3x} \, dx = x \cdot \frac{1}{3} e^{3x} - \int \frac{1}{3} e^{3x} \, dx \]
\[ = \frac{1}{3} x e^{3x} - \frac{1}{9} e^{3x} \]

5. **Substituting Back**:
\[ \int x^2 e^{3x} \, dx = \frac{1}{3} x^2 e^{3x} - \frac{2}{3} \left( \frac{1}{3} x e^{3x} - \frac{1}{9} e^{3x} \right) \]
\[ = \frac{1}{3} x^2 e^{3x} - \frac{2}{9} x e^{3x} + \frac{2}{27} e^{3x} + C \]

### Final Answer for Part (ii):
\[ \int x^2 e^{3x} \, dx = \frac{1}{3} x^2 e^{3x} - \frac{2}{9} x e^{3x} + \frac{2}{27} e^{3x} + C \]

---