If `int(cos^(4)x)/(sin^(2)x)dx=A cot x +B sin 2x +(C)/(2)x+D`, then

Options

A. `A=-2 , B=1//4`
B. `B=-1//4, C=-3`
C. `B=1//4 , C =-3`
D. none of these

Explanation

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

### Step 1: Rewrite the integral
We start with the integral:
\[ \int \frac{\cos^4 x}{\sin^2 x} \, dx \]
We can rewrite \(\cos^4 x\) using the identity \(\cos^2 x = 1 - \sin^2 x\):
\[ \cos^4 x = (\cos^2 x)^2 = (1 - \sin^2 x)^2 \]
Thus, we have:
\[ \int \frac{(1 - \sin^2 x)^2}{\sin^2 x} \, dx \]

### Step 2: Expand the integrand
Expanding \((1 - \sin^2 x)^2\):
\[ (1 - \sin^2 x)^2 = 1 - 2\sin^2 x + \sin^4 x \]
So, the integral becomes:
\[ \int \left( \frac{1}{\sin^2 x} - 2 + \frac{\sin^4 x}{\sin^2 x} \right) \, dx = \int \left( \csc^2 x - 2 + \sin^2 x \right) \, dx \]

### Step 3: Separate the integral
Now we can separate the integral:
\[ \int \csc^2 x \, dx - 2 \int dx + \int \sin^2 x \, dx \]

### Step 4: Calculate each integral
1. The integral of \(\csc^2 x\):
\[ \int \csc^2 x \, dx = -\cot x \]

2. The integral of \(-2\):
\[ -2 \int dx = -2x \]

3. The integral of \(\sin^2 x\):
We can use the identity \(\sin^2 x = \frac{1 - \cos 2x}{2}\):
\[ \int \sin^2 x \, dx = \int \frac{1 - \cos 2x}{2} \, dx = \frac{1}{2} \int dx - \frac{1}{2} \int \cos 2x \, dx \]
\[ = \frac{x}{2} - \frac{1}{4} \sin 2x \]

### Step 5: Combine the results
Now we combine the results of the integrals:
\[ -\cot x - 2x + \left(\frac{x}{2} - \frac{1}{4} \sin 2x\right) \]
Combining terms:
\[ -\cot x - 2x + \frac{x}{2} - \frac{1}{4} \sin 2x = -\cot x - \frac{4x}{2} + \frac{x}{2} - \frac{1}{4} \sin 2x \]
\[ = -\cot x - \frac{3x}{2} - \frac{1}{4} \sin 2x \]

### Step 6: Final expression
Thus, the integral can be expressed as:
\[ -\cot x - \frac{1}{4} \sin 2x - \frac{3}{2} x + C \]

### Step 7: Identify coefficients
From the expression, we can identify:
- \(A = -1\)
- \(B = -\frac{1}{4}\)
- \(C = -3\)
- \(D = C\) (constant of integration)

### Summary of Coefficients
- \(A = -1\)
- \(B = -\frac{1}{4}\)
- \(C = -3\)
- \(D\) is a constant.