If `intcos^(4)xdx=Ax +B sin 2x+C sin 4x+D`, then `{A,B,C}` equals to

Options

A. `{(3)/(8),(1)/(32),(1)/(4)}`
B. `{(3)/(8),(1)/(4),(1)/(32)}`
C. `{(1)/(32),(1)/(4),(3)/(8)}`
D. `{(1)/(4),(3)/(8),(1)/(32)}`

Explanation

To solve the integral \( \int \cos^4 x \, dx \) and express it in the form \( Ax + B \sin 2x + C \sin 4x + D \), we will follow these steps:

### Step 1: Rewrite \( \cos^4 x \)
We start by using the identity for \( \cos^2 x \):
\[ \cos^2 x = \frac{1 + \cos 2x}{2} \]
Thus,
\[ \cos^4 x = \left( \cos^2 x \right)^2 = \left( \frac{1 + \cos 2x}{2} \right)^2 = \frac{(1 + \cos 2x)^2}{4} \]

### Step 2: Expand \( \cos^4 x \)
Now we expand \( (1 + \cos 2x)^2 \):
\[ (1 + \cos 2x)^2 = 1 + 2\cos 2x + \cos^2 2x \]
Next, we substitute \( \cos^2 2x \) using the identity again:
\[ \cos^2 2x = \frac{1 + \cos 4x}{2} \]
So,
\[ (1 + \cos 2x)^2 = 1 + 2\cos 2x + \frac{1 + \cos 4x}{2} = 1 + 2\cos 2x + \frac{1}{2} + \frac{1}{2}\cos 4x = \frac{3}{2} + 2\cos 2x + \frac{1}{2}\cos 4x \]

### Step 3: Substitute back into the integral
Now substituting back into the integral:
\[ \int \cos^4 x \, dx = \int \frac{1}{4} \left( \frac{3}{2} + 2\cos 2x + \frac{1}{2}\cos 4x \right) \, dx \]
This simplifies to:
\[ \int \cos^4 x \, dx = \frac{1}{4} \int \left( \frac{3}{2} + 2\cos 2x + \frac{1}{2}\cos 4x \right) \, dx \]

### Step 4: Integrate each term
Now we integrate each term separately:
1. \( \int \frac{3}{2} \, dx = \frac{3}{2}x \)
2. \( \int 2\cos 2x \, dx = 2 \cdot \frac{\sin 2x}{2} = \sin 2x \)
3. \( \int \frac{1}{2}\cos 4x \, dx = \frac{1}{2} \cdot \frac{\sin 4x}{4} = \frac{1}{8}\sin 4x \)

Putting it all together:
\[ \int \cos^4 x \, dx = \frac{1}{4} \left( \frac{3}{2}x + \sin 2x + \frac{1}{8}\sin 4x \right) + C \]
\[ = \frac{3}{8}x + \frac{1}{4}\sin 2x + \frac{1}{32}\sin 4x + C \]

### Step 5: Identify coefficients
From the expression \( Ax + B \sin 2x + C \sin 4x + D \), we can identify:
- \( A = \frac{3}{8} \)
- \( B = \frac{1}{4} \)
- \( C = \frac{1}{32} \)

### Final Answer
Thus, the values of \( \{A, B, C\} \) are:
\[ \{A, B, C\} = \left\{ \frac{3}{8}, \frac{1}{4}, \frac{1}{32} \right\} \]