The value of `int(ln(cotx))/(sin2x)dx` is equal to (where, C is the constant of integration)

Options

A. `((ln(cotx))^(2))/(2)+C`
B. `((ln(cotx))^(2))/(4)+C`
C. `((ln(cotx))^(2))/(6)+C`
D. `-(1)/(4)(ln(cotx))^(2)+C`

Answer

To solve the integral \( I = \int \frac{\ln(\cot x)}{\sin(2x)} \, dx \), we will use integration by parts and some trigonometric identities. Here’s a step-by-step solution:

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int \frac{\ln(\cot x)}{\sin(2x)} \, dx \]
Using the identity \( \sin(2x) = 2 \sin(x) \cos(x) \), we can rewrite the integral as:
\[ I = \int \frac{\ln(\cot x)}{2 \sin(x) \cos(x)} \, dx \]

### Step 2: Apply Integration by Parts
We will apply integration by parts, which states:
\[ \int u \, dv = uv - \int v \, du \]
Let:
- \( u = \ln(\cot x) \) → \( du = -\csc^2(x) \, dx \)
- \( dv = \frac{1}{\sin(2x)} \, dx = \frac{1}{2 \sin(x) \cos(x)} \, dx \)

Now, we need to find \( v \):
\[ v = \int \frac{1}{\sin(2x)} \, dx = \frac{1}{2} \ln|\tan(x)| + C \]

### Step 3: Substitute into Integration by Parts Formula
Substituting \( u \), \( du \), and \( v \) into the integration by parts formula:
\[ I = \ln(\cot x) \cdot \left(\frac{1}{2} \ln|\tan(x)|\right) - \int \left(\frac{1}{2} \ln|\tan(x)|\right)(-\csc^2(x)) \, dx \]
This simplifies to:
\[ I = \frac{1}{2} \ln(\cot x) \ln|\tan(x)| + \frac{1}{2} \int \ln|\tan(x)| \csc^2(x) \, dx \]

### Step 4: Solve the Remaining Integral
Now we need to evaluate the integral:
\[ \int \ln|\tan(x)| \csc^2(x) \, dx \]
Using the substitution \( \tan(x) = t \), we have \( \sec^2(x) \, dx = dt \), and thus:
\[ \int \ln|t| \, dt = t \ln|t| - t + C = \tan(x) \ln|\tan(x)| - \tan(x) + C \]

### Step 5: Combine Results
Now substitute back into our expression for \( I \):
\[ I = \frac{1}{2} \ln(\cot x) \ln|\tan(x)| + \frac{1}{2} \left( \tan(x) \ln|\tan(x)| - \tan(x) \right) + C \]

### Final Result
Thus, the final result for the integral is:
\[ I = \frac{1}{2} \ln(\cot x) \ln|\tan(x)| + \frac{1}{2} \tan(x) \ln|\tan(x)| - \frac{1}{2} \tan(x) + C \]