`int e^x cos 3x dx`

Answer

To solve the integral \( \int e^x \cos(3x) \, dx \), we will use the method of integration by parts.

### Step 1: Set up the integral
Let:
\[ I = \int e^x \cos(3x) \, dx \]

### Step 2: Apply integration by parts
We choose:
- \( u = e^x \) (thus \( du = e^x \, dx \))
- \( dv = \cos(3x) \, dx \) (thus \( v = \frac{1}{3} \sin(3x) \))

Using the integration by parts formula:
\[ \int u \, dv = uv - \int v \, du \]
we have:
\[ I = e^x \cdot \frac{1}{3} \sin(3x) - \int \frac{1}{3} \sin(3x) e^x \, dx \]
This simplifies to:
\[ I = \frac{1}{3} e^x \sin(3x) - \frac{1}{3} \int e^x \sin(3x) \, dx \]

### Step 3: Set up the new integral
Let:
\[ J = \int e^x \sin(3x) \, dx \]
Then we can rewrite \( I \):
\[ I = \frac{1}{3} e^x \sin(3x) - \frac{1}{3} J \]

### Step 4: Apply integration by parts to \( J \)
Now, we will apply integration by parts to \( J \):
- \( u = e^x \) (thus \( du = e^x \, dx \))
- \( dv = \sin(3x) \, dx \) (thus \( v = -\frac{1}{3} \cos(3x) \))

Using the integration by parts formula again:
\[ J = e^x \cdot \left(-\frac{1}{3} \cos(3x)\right) - \int -\frac{1}{3} \cos(3x) e^x \, dx \]
This simplifies to:
\[ J = -\frac{1}{3} e^x \cos(3x) + \frac{1}{3} \int e^x \cos(3x) \, dx \]
Thus:
\[ J = -\frac{1}{3} e^x \cos(3x) + \frac{1}{3} I \]

### Step 5: Substitute \( J \) back into \( I \)
Now substitute \( J \) back into the equation for \( I \):
\[ I = \frac{1}{3} e^x \sin(3x) - \frac{1}{3} \left( -\frac{1}{3} e^x \cos(3x) + \frac{1}{3} I \right) \]
This simplifies to:
\[ I = \frac{1}{3} e^x \sin(3x) + \frac{1}{9} e^x \cos(3x) - \frac{1}{9} I \]

### Step 6: Solve for \( I \)
Now, combine like terms:
\[ I + \frac{1}{9} I = \frac{1}{3} e^x \sin(3x) + \frac{1}{9} e^x \cos(3x) \]
\[ \frac{10}{9} I = \frac{1}{3} e^x \sin(3x) + \frac{1}{9} e^x \cos(3x) \]
Multiply both sides by \( \frac{9}{10} \):
\[ I = \frac{3}{10} e^x \sin(3x) + \frac{1}{10} e^x \cos(3x) \]

### Step 7: Final answer
Thus, the integral is:
\[ \int e^x \cos(3x) \, dx = \frac{1}{10} e^x (3 \sin(3x) + \cos(3x)) + C \]
where \( C \) is the constant of integration.