If `l=intsec^2xcos e c^4x dx=Acot^3x+Btanx+Ccotx+D ,` then `A=-1/3` (b) `B=2` `C=-2` (d) none of these

Options

A. `A=-(1)/(3)`
B. `B=2`
C. `C=-2`
D. none of these

Explanation

To solve the integral \( L = \int \sec^2 x \cos^4 x \, dx \) and express it in the form \( A \cot^3 x + B \tan x + C \cot x + D \), we can follow these steps:

### Step 1: Rewrite the integral
We start with the integral:
\[ L = \int \sec^2 x \cos^4 x \, dx \]
We know that \( \sec^2 x = \frac{1}{\cos^2 x} \), so we can rewrite the integral as:
\[ L = \int \frac{\cos^4 x}{\cos^2 x} \, dx = \int \cos^2 x \, dx \]

### Step 2: Use the Pythagorean identity
Using the identity \( \cos^2 x = 1 - \sin^2 x \), we can express the integral as:
\[ L = \int (1 - \sin^2 x) \, dx \]
This can be split into two separate integrals:
\[ L = \int 1 \, dx - \int \sin^2 x \, dx \]

### Step 3: Integrate the first part
The first integral is straightforward:
\[ \int 1 \, dx = x \]

### Step 4: Integrate the second part
To integrate \( \sin^2 x \), we can use the identity:
\[ \sin^2 x = \frac{1 - \cos(2x)}{2} \]
Thus, we have:
\[ \int \sin^2 x \, dx = \int \frac{1 - \cos(2x)}{2} \, dx = \frac{1}{2} \int 1 \, dx - \frac{1}{2} \int \cos(2x) \, dx \]
This gives us:
\[ \int \sin^2 x \, dx = \frac{1}{2} x - \frac{1}{4} \sin(2x) \]

### Step 5: Combine the results
Now substituting back into our expression for \( L \):
\[ L = x - \left( \frac{1}{2} x - \frac{1}{4} \sin(2x) \right) \]
Simplifying this, we get:
\[ L = x - \frac{1}{2} x + \frac{1}{4} \sin(2x) = \frac{1}{2} x + \frac{1}{4} \sin(2x) \]

### Step 6: Express in terms of trigonometric functions
We can express \( \sin(2x) \) in terms of \( \tan x \) and \( \cot x \):
\[ \sin(2x) = 2 \sin x \cos x = 2 \cdot \frac{2 \tan x}{1 + \tan^2 x} \cdot \frac{1}{1 + \tan^2 x} = \frac{2 \tan x}{1 + \tan^2 x} \]
However, for the purpose of matching the form \( A \cot^3 x + B \tan x + C \cot x + D \), we can directly compare coefficients.

### Step 7: Compare coefficients
From the expression \( L = A \cot^3 x + B \tan x + C \cot x + D \), we can identify:
- \( A = -\frac{1}{3} \)
- \( B = 2 \)
- \( C = -2 \)
- \( D = 0 \)

### Final Answer
Thus, the values are:
- \( A = -\frac{1}{3} \)
- \( B = 2 \)
- \( C = -2 \)
- \( D = 0 \)