The value of `int_(0)^(pi//2) cosec(x-pi//3)cosec(x-pi//6)dx` is

Options

A. 2 log 3
B. `-2 log 3`
C. log 3
D. none of these

Answer

To solve the integral \( I = \int_{0}^{\frac{\pi}{2}} \csc\left(x - \frac{\pi}{3}\right) \csc\left(x - \frac{\pi}{6}\right) \, dx \), we will follow these steps:

### Step 1: Rewrite the Cosecant Function
We start by rewriting the cosecant functions in terms of sine:
\[ I = \int_{0}^{\frac{\pi}{2}} \frac{1}{\sin\left(x - \frac{\pi}{3}\right)} \cdot \frac{1}{\sin\left(x - \frac{\pi}{6}\right)} \, dx \]

### Step 2: Multiply and Divide by a Constant
Next, we multiply and divide the integrand by \(\sin\left(\frac{\pi}{3} - \frac{\pi}{6}\right)\):
\[ I = \int_{0}^{\frac{\pi}{2}} \frac{\sin\left(\frac{\pi}{3} - \frac{\pi}{6}\right)}{\sin\left(x - \frac{\pi}{3}\right) \sin\left(x - \frac{\pi}{6}\right)} \, dx \]

### Step 3: Simplify the Constant
Calculate \(\sin\left(\frac{\pi}{3} - \frac{\pi}{6}\right)\):
\[ \sin\left(\frac{\pi}{3} - \frac{\pi}{6}\right) = \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \]
Thus, we have:
\[ I = 2 \int_{0}^{\frac{\pi}{2}} \frac{1}{\sin\left(x - \frac{\pi}{3}\right) \sin\left(x - \frac{\pi}{6}\right)} \, dx \]

### Step 4: Use the Sine Difference Formula
Using the sine difference formula:
\[ \sin(a - b) = \sin a \cos b - \cos a \sin b \]
we can express the integrand:
\[ I = 2 \int_{0}^{\frac{\pi}{2}} \frac{\sin\left(x - \frac{\pi}{6}\right) \cos\left(x - \frac{\pi}{3}\right) - \cos\left(x - \frac{\pi}{6}\right) \sin\left(x - \frac{\pi}{3}\right)}{\sin\left(x - \frac{\pi}{3}\right) \sin\left(x - \frac{\pi}{6}\right)} \, dx \]

### Step 5: Simplify the Integral
The integral can be simplified further:
\[ I = 2 \int_{0}^{\frac{\pi}{2}} \left(\cot\left(x - \frac{\pi}{3}\right) - \cot\left(x - \frac{\pi}{6}\right)\right) \, dx \]

### Step 6: Integrate
The integral of \(\cot x\) is:
\[ \int \cot x \, dx = \log(\sin x) \]
Thus, we have:
\[ I = 2 \left[ \log(\sin\left(x - \frac{\pi}{3}\right)) - \log(\sin\left(x - \frac{\pi}{6}\right)) \right]_{0}^{\frac{\pi}{2}} \]

### Step 7: Evaluate the Limits
Evaluate the limits:
\[ = 2 \left[ \log(\sin\left(\frac{\pi}{2} - \frac{\pi}{3}\right)) - \log(\sin\left(\frac{\pi}{2} - \frac{\pi}{6}\right)) - \left(\log(\sin\left(-\frac{\pi}{3}\right)) - \log(\sin\left(-\frac{\pi}{6}\right))\right) \right] \]
Calculating the sine values:
\[ = 2 \left[ \log\left(\frac{1}{2}\right) - \log\left(\frac{\sqrt{3}}{2}\right) + \log\left(\frac{\sqrt{3}}{2}\right) - \log\left(\frac{1}{2}\right) \right] \]

### Step 8: Combine the Logarithms
Using properties of logarithms:
\[ = 2 \left[ \log\left(\frac{1/2}{\sqrt{3}/2}\right) \right] = 2 \log\left(\frac{1}{\sqrt{3}}\right) = -2 \log(\sqrt{3}) = -2 \cdot \frac{1}{2} \log(3) = -2 \log(3) \]

### Final Answer
Thus, the value of the integral is:
\[ \boxed{-2 \log 3} \]