What is `intsinxlog(tanx)dx` equal to ?

Options

A. `cosxlogtanx+logtan(x//2)+c`
B. `-cosxlogtanx+logtan(x//2)+c`
C. `cosxlogtanx+logcot(x//2)+c`
D. `-cosxlogtanx+logcot(x//2)+c`

Answer

To solve the integral \( \int \sin x \log(\tan x) \, dx \), we will use the integration by parts formula. The formula states:

\[ \int u \, dv = uv - \int v \, du \]

### Step 1: Choose \( u \) and \( dv \)

We will choose:
- \( u = \log(\tan x) \) (logarithmic function)
- \( dv = \sin x \, dx \) (trigonometric function)

### Step 2: Differentiate \( u \) and integrate \( dv \)

Now we need to find \( du \) and \( v \):
- Differentiate \( u \):
\[ du = \frac{d}{dx}(\log(\tan x)) \, dx = \frac{1}{\tan x} \cdot \sec^2 x \, dx = \frac{\sec^2 x}{\tan x} \, dx \]

- Integrate \( dv \):
\[ v = \int \sin x \, dx = -\cos x \]

### Step 3: Apply the integration by parts formula

Now we can substitute \( u \), \( v \), \( du \), and \( dv \) into the integration by parts formula:
\[ \int \sin x \log(\tan x) \, dx = -\cos x \log(\tan x) - \int -\cos x \cdot \frac{\sec^2 x}{\tan x} \, dx \]

### Step 4: Simplify the integral

The integral becomes:
\[ -\cos x \log(\tan x) + \int \frac{\cos x \sec^2 x}{\tan x} \, dx \]
Since \( \sec^2 x = \frac{1}{\cos^2 x} \) and \( \tan x = \frac{\sin x}{\cos x} \), we can rewrite:
\[ \frac{\cos x \sec^2 x}{\tan x} = \frac{\cos x \cdot \frac{1}{\cos^2 x}}{\frac{\sin x}{\cos x}} = \frac{1}{\sin x \cos x} \]

Thus, the integral simplifies to:
\[ -\cos x \log(\tan x) + \int \frac{1}{\sin x \cos x} \, dx \]

### Step 5: Solve the remaining integral

The integral \( \int \frac{1}{\sin x \cos x} \, dx \) can be rewritten as:
\[ \int \frac{2}{\sin(2x)} \, dx = 2 \int \csc(2x) \, dx \]
The integral of \( \csc(2x) \) is:
\[ - \frac{1}{2} \log \left| \csc(2x) + \cot(2x) \right| + C \]

### Step 6: Combine results

Putting everything together, we have:
\[ \int \sin x \log(\tan x) \, dx = -\cos x \log(\tan x) - \log \left| \csc(2x) + \cot(2x) \right| + C \]

### Final Result

Thus, the final result is:
\[ \int \sin x \log(\tan x) \, dx = -\cos x \log(\tan x) + \log \left| \tan \left( \frac{x}{2} \right) \right| + C \]