(i) `int e^(3x) cos 5x dx`
(ii) `int e^(3x) sin 4x dx`

Answer

To solve the integrals given in the question, we will use the standard formulas for the integration of the product of an exponential function and trigonometric functions.

### (i) Integral of \( e^{3x} \cos(5x) \, dx \)

1. **Identify the parameters:**
- Here, \( a = 3 \) and \( b = 5 \).

2. **Use the formula for integration:**
\[ \int e^{ax} \cos(bx) \, dx = \frac{e^{ax}}{a^2 + b^2} (a \cos(bx) + b \sin(bx)) + C \]
Substituting \( a \) and \( b \):
\[ \int e^{3x} \cos(5x) \, dx = \frac{e^{3x}}{3^2 + 5^2} (3 \cos(5x) + 5 \sin(5x)) + C \]

3. **Calculate \( a^2 + b^2 \):**
\[ 3^2 + 5^2 = 9 + 25 = 34 \]

4. **Substitute back into the formula:**
\[ \int e^{3x} \cos(5x) \, dx = \frac{e^{3x}}{34} (3 \cos(5x) + 5 \sin(5x)) + C \]

### Final Result for (i):
\[ \int e^{3x} \cos(5x) \, dx = \frac{e^{3x}}{34} (3 \cos(5x) + 5 \sin(5x)) + C \]

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### (ii) Integral of \( e^{3x} \sin(4x) \, dx \)

1. **Identify the parameters:**
- Here, \( a = 3 \) and \( b = 4 \).

2. **Use the formula for integration:**
\[ \int e^{ax} \sin(bx) \, dx = \frac{e^{ax}}{a^2 + b^2} (a \sin(bx) - b \cos(bx)) + C \]
Substituting \( a \) and \( b \):
\[ \int e^{3x} \sin(4x) \, dx = \frac{e^{3x}}{3^2 + 4^2} (3 \sin(4x) - 4 \cos(4x)) + C \]

3. **Calculate \( a^2 + b^2 \):**
\[ 3^2 + 4^2 = 9 + 16 = 25 \]

4. **Substitute back into the formula:**
\[ \int e^{3x} \sin(4x) \, dx = \frac{e^{3x}}{25} (3 \sin(4x) - 4 \cos(4x)) + C \]

### Final Result for (ii):
\[ \int e^{3x} \sin(4x) \, dx = \frac{e^{3x}}{25} (3 \sin(4x) - 4 \cos(4x)) + C \]

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