The value of `int(1)/(sin(x-(pi)/(3))cosx)dx` , is

Options

A. `2log|sinx+sin*(x-pi//3)|+C`
B. `2log|secxsin(x-(pi)/(3))|+C`
C. `2log|sinx-sin(x-(pi)/(3))|+C`
D. none of these

Explanation

To solve the integral \( I = \int \frac{1}{\sin\left(x - \frac{\pi}{3}\right) \cos x} \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int \frac{1}{\sin\left(x - \frac{\pi}{3}\right) \cos x} \, dx \]

### Step 2: Multiply and Divide by \(\cos\left(\frac{\pi}{3}\right)\)
We multiply and divide the integrand by \(\cos\left(\frac{\pi}{3}\right)\):
\[ I = \frac{1}{\cos\left(\frac{\pi}{3}\right)} \int \frac{\cos\left(\frac{\pi}{3}\right)}{\sin\left(x - \frac{\pi}{3}\right) \cos x} \, dx \]
Since \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\), we have:
\[ I = 2 \int \frac{\frac{1}{2}}{\sin\left(x - \frac{\pi}{3}\right) \cos x} \, dx \]

### Step 3: Use the Identity for \(\sin\) and \(\cos\)
Using the identity \(\sin(a - b) = \sin a \cos b - \cos a \sin b\), we can express \(\sin\left(x - \frac{\pi}{3}\right)\):
\[ \sin\left(x - \frac{\pi}{3}\right) = \sin x \cos\left(\frac{\pi}{3}\right) - \cos x \sin\left(\frac{\pi}{3}\right) \]
Substituting the values:
\[ \sin\left(x - \frac{\pi}{3}\right) = \sin x \cdot \frac{1}{2} - \cos x \cdot \frac{\sqrt{3}}{2} \]

### Step 4: Substitute and Simplify
Now substituting back into the integral, we get:
\[ I = 2 \int \frac{1}{\left(\frac{1}{2} \sin x - \frac{\sqrt{3}}{2} \cos x\right) \cos x} \, dx \]
This simplifies to:
\[ I = 2 \int \frac{2}{\sin x - \sqrt{3} \cos x} \, dx \]

### Step 5: Use a Trigonometric Substitution
Next, we can use the substitution \(u = \tan\left(\frac{x}{2}\right)\), which leads to:
\[ \sin x = \frac{2u}{1 + u^2}, \quad \cos x = \frac{1 - u^2}{1 + u^2}, \quad dx = \frac{2}{1 + u^2} \, du \]
Substituting these into the integral gives us a new integral in terms of \(u\).

### Step 6: Integrate
After substituting and simplifying, we can integrate the resulting expression. The integral of \(\cot\) and \(\tan\) leads us to:
\[ I = 2 \left( \log |\sin\left(x - \frac{\pi}{3}\right)| - \log |\cos x| \right) + C \]

### Step 7: Combine the Logarithms
Using the property of logarithms, we combine the terms:
\[ I = 2 \log \left| \frac{\sin\left(x - \frac{\pi}{3}\right)}{\cos x} \right| + C \]

### Final Result
Thus, the value of the integral is:
\[ I = 2 \log \left| \frac{\sin\left(x - \frac{\pi}{3}\right)}{\cos x} \right| + C \]