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

Answer

To solve the integrals step by step, we will use the method of integration by parts, which is based on the formula:

\[ \int u \, dv = uv - \int v \, du \]

where \( u \) is a function we differentiate, and \( dv \) is a function we integrate.

### (i) \(\int x^2 e^x \, dx\)

1. **Choose \( u \) and \( dv \)**:
- Let \( u = x^2 \) (then \( du = 2x \, dx \))
- Let \( dv = e^x \, dx \) (then \( v = e^x \))

2. **Apply integration by parts**:
\[ \int x^2 e^x \, dx = x^2 e^x - \int e^x (2x) \, dx \]

3. **Now we need to solve \(\int 2x e^x \, dx\)** using integration by parts again:
- Let \( u = 2x \) (then \( du = 2 \, dx \))
- Let \( dv = e^x \, dx \) (then \( v = e^x \))

4. **Apply integration by parts again**:
\[ \int 2x e^x \, dx = 2x e^x - \int 2 e^x \, dx = 2x e^x - 2 e^x \]

5. **Substituting back into the first equation**:
\[ \int x^2 e^x \, dx = x^2 e^x - (2x e^x - 2 e^x) = x^2 e^x - 2x e^x + 2 e^x + C \]

6. **Final result**:
\[ \int x^2 e^x \, dx = e^x (x^2 - 2x + 2) + C \]

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

1. **Choose \( u \) and \( dv \)**:
- Let \( u = x^2 \) (then \( du = 2x \, dx \))
- Let \( dv = e^{3x} \, dx \) (then \( v = \frac{1}{3} e^{3x} \))

2. **Apply integration by parts**:
\[ \int x^2 e^{3x} \, dx = \frac{1}{3} x^2 e^{3x} - \int \frac{1}{3} e^{3x} (2x) \, dx \]

3. **Now we need to solve \(\int 2x e^{3x} \, dx\)** using integration by parts again:
- Let \( u = 2x \) (then \( du = 2 \, dx \))
- Let \( dv = e^{3x} \, dx \) (then \( v = \frac{1}{3} e^{3x} \))

4. **Apply integration by parts again**:
\[ \int 2x e^{3x} \, dx = \frac{2}{3} x e^{3x} - \int \frac{2}{3} e^{3x} \, dx = \frac{2}{3} x e^{3x} - \frac{2}{9} e^{3x} \]

5. **Substituting back into the first equation**:
\[ \int x^2 e^{3x} \, dx = \frac{1}{3} x^2 e^{3x} - \left( \frac{2}{3} x e^{3x} - \frac{2}{9} e^{3x} \right) \]

6. **Final result**:
\[ \int x^2 e^{3x} \, dx = \frac{1}{3} x^2 e^{3x} - \frac{2}{3} x e^{3x} + \frac{2}{9} e^{3x} + C \]

### (iii) \(\int x^3 e^x \, dx\)

1. **Choose \( u \) and \( dv \)**:
- Let \( u = x^3 \) (then \( du = 3x^2 \, dx \))
- Let \( dv = e^x \, dx \) (then \( v = e^x \))

2. **Apply integration by parts**:
\[ \int x^3 e^x \, dx = x^3 e^x - \int e^x (3x^2) \, dx \]

3. **Now we need to solve \(\int 3x^2 e^x \, dx\)** using the result from part (i):
\[ \int 3x^2 e^x \, dx = 3 \left( e^x (x^2 - 2x + 2) \right) = 3e^x (x^2 - 2x + 2) \]

4. **Substituting back into the first equation**:
\[ \int x^3 e^x \, dx = x^3 e^x - 3e^x (x^2 - 2x + 2) \]

5. **Final result**:
\[ \int x^3 e^x \, dx = e^x (x^3 - 3x^2 + 6x - 6) + C \]