`int" cos"^(4) " 2x dx "`

Answer

To solve the integral \( \int \cos^4(2x) \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We start by rewriting \( \cos^4(2x) \) as \( (\cos^2(2x))^2 \):
\[ \int \cos^4(2x) \, dx = \int (\cos^2(2x))^2 \, dx \]

### Step 2: Use the Identity for Cosine Squared
We can use the identity \( \cos^2(x) = \frac{1 + \cos(2x)}{2} \) to express \( \cos^2(2x) \):
\[ \cos^2(2x) = \frac{1 + \cos(4x)}{2} \]
Thus,
\[ \cos^4(2x) = \left(\frac{1 + \cos(4x)}{2}\right)^2 \]

### Step 3: Expand the Expression
Now we expand the square:
\[ \cos^4(2x) = \frac{(1 + \cos(4x))^2}{4} = \frac{1 + 2\cos(4x) + \cos^2(4x)}{4} \]

### Step 4: Substitute Back into the Integral
Now substitute this back into the integral:
\[ \int \cos^4(2x) \, dx = \int \frac{1 + 2\cos(4x) + \cos^2(4x)}{4} \, dx \]
This simplifies to:
\[ \frac{1}{4} \int (1 + 2\cos(4x) + \cos^2(4x)) \, dx \]

### Step 5: Break Down the Integral
We can break this down into three separate integrals:
\[ \frac{1}{4} \left( \int 1 \, dx + 2 \int \cos(4x) \, dx + \int \cos^2(4x) \, dx \right) \]

### Step 6: Calculate Each Integral
1. The first integral:
\[ \int 1 \, dx = x \]

2. The second integral:
\[ \int \cos(4x) \, dx = \frac{\sin(4x)}{4} \]

3. For the third integral \( \int \cos^2(4x) \, dx \), we use the identity again:
\[ \cos^2(4x) = \frac{1 + \cos(8x)}{2} \]
Thus,
\[ \int \cos^2(4x) \, dx = \int \frac{1 + \cos(8x)}{2} \, dx = \frac{1}{2} \left( x + \frac{\sin(8x)}{8} \right) \]

### Step 7: Combine the Results
Now, substituting back into our expression:
\[ \frac{1}{4} \left( x + 2 \cdot \frac{\sin(4x)}{4} + \frac{1}{2} \left( x + \frac{\sin(8x)}{8} \right) \right) \]
This simplifies to:
\[ \frac{1}{4} \left( x + \frac{\sin(4x)}{2} + \frac{x}{2} + \frac{\sin(8x)}{16} \right) \]
Combining the \( x \) terms:
\[ \frac{1}{4} \left( \frac{3x}{2} + \frac{\sin(4x)}{2} + \frac{\sin(8x)}{16} \right) \]

### Final Step: Write the Final Answer
Thus, the final answer is:
\[ \frac{3x}{8} + \frac{\sin(4x)}{8} + \frac{\sin(8x)}{64} + C \]