If `int(cos^(2)x)/(sin^(6)x)dx=A cot^(5)x+B cot^(3)x+k`, then `A+B` equals

Options

A. `(8)/(15)`
B. `-(8)/(15)`
C. `(15)/(8)`
D. `-(15)/(8)`

Answer

To solve the integral \(\int \frac{\cos^2 x}{\sin^6 x} \, dx\) and express it in the form \(A \cot^5 x + B \cot^3 x + k\), we will follow these steps:

### Step 1: Rewrite the integral
We start with the integral:
\[ \int \frac{\cos^2 x}{\sin^6 x} \, dx \]
We can rewrite \(\cos^2 x\) as \(\frac{1}{\sin^2 x} - \frac{1}{\tan^2 x}\) to facilitate the integration.

### Step 2: Use trigonometric identities
Using the identity \(\cot x = \frac{\cos x}{\sin x}\), we can express \(\cos^2 x\) in terms of \(\cot x\):
\[ \cos^2 x = \frac{1}{\sin^2 x} \cdot \sin^2 x = \cot^2 x \cdot \sin^4 x \]
Thus, we can rewrite the integral as:
\[ \int \frac{\cot^2 x}{\sin^4 x} \, dx \]

### Step 3: Change variables
Let \(t = \tan x\). Then, \(dt = \sec^2 x \, dx\) or \(dx = \frac{dt}{1 + t^2}\). Also, \(\sin x = \frac{t}{\sqrt{1+t^2}}\) and \(\cos x = \frac{1}{\sqrt{1+t^2}}\). Therefore, we can express \(\sin^6 x\) and \(\cos^2 x\) in terms of \(t\):
\[ \sin^6 x = \left(\frac{t}{\sqrt{1+t^2}}\right)^6 = \frac{t^6}{(1+t^2)^3} \]
\[ \cos^2 x = \frac{1}{1+t^2} \]

### Step 4: Substitute into the integral
Substituting these into the integral gives:
\[ \int \frac{\frac{1}{1+t^2}}{\frac{t^6}{(1+t^2)^3}} \cdot \frac{dt}{1+t^2} = \int \frac{(1+t^2)^2}{t^6} \, dt \]

### Step 5: Simplify the integral
This simplifies to:
\[ \int \left(\frac{1}{t^6} + \frac{2}{t^4} + \frac{1}{t^2}\right) \, dt \]

### Step 6: Integrate term by term
Now we integrate each term:
1. \(\int t^{-6} \, dt = -\frac{1}{5t^5}\)
2. \(\int 2t^{-4} \, dt = -\frac{2}{3t^3}\)
3. \(\int t^{-2} \, dt = -\frac{1}{t}\)

Putting it all together:
\[ -\frac{1}{5t^5} - \frac{2}{3t^3} - \frac{1}{t} + C \]

### Step 7: Substitute back for \(t\)
Recall that \(t = \tan x\), so we substitute back:
\[ -\frac{1}{5 \tan^5 x} - \frac{2}{3 \tan^3 x} - \frac{1}{\tan x} + C \]

### Step 8: Express in terms of cotangent
Using \(\cot x = \frac{1}{\tan x}\), we rewrite the expression:
\[ -\frac{1}{5 \cot^5 x} - \frac{2}{3 \cot^3 x} - \cot x + C \]

### Step 9: Identify coefficients
From the expression, we can identify:
- \(A = -\frac{1}{5}\)
- \(B = -\frac{2}{3}\)

### Step 10: Calculate \(A + B\)
Now, we calculate:
\[ A + B = -\frac{1}{5} - \frac{2}{3} \]
To add these fractions, we find a common denominator (15):
\[ A + B = -\frac{3}{15} - \frac{10}{15} = -\frac{13}{15} \]

Thus, the final answer is:
\[ A + B = -\frac{13}{15} \]