The value of `int_(pi//6)^(pi//3) (1)/( sin 2x) dx` is

Options

A. `log 3`
B. `(1)/(2) log 2`
C. `(1)/(2) log 3`
D. none of these

Answer

To solve the integral \( I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{1}{\sin 2x} \, dx \), we will follow these steps:

### Step 1: Rewrite the Integral
We start by rewriting the integral using the identity for sine:
\[ \sin 2x = 2 \sin x \cos x \]
Thus, we can express the integral as:
\[ I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{1}{\sin 2x} \, dx = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{1}{2 \sin x \cos x} \, dx \]

### Step 2: Simplify the Integral
Next, we can simplify the integral:
\[ I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{1}{2 \sin x \cos x} \, dx = \frac{1}{2} \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{1}{\sin x \cos x} \, dx \]
Using the identity \( \frac{1}{\sin x \cos x} = \frac{2}{\sin 2x} \), we can rewrite the integral:
\[ I = \frac{1}{2} \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{2}{\sin 2x} \, dx = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{1}{\sin 2x} \, dx \]

### Step 3: Substitution
Now we will perform a substitution. Let \( t = \tan x \), then \( dt = \sec^2 x \, dx \) or \( dx = \frac{dt}{\sec^2 x} = \frac{dt}{1 + t^2} \).
The limits change as follows:
- When \( x = \frac{\pi}{6} \), \( t = \tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}} \)
- When \( x = \frac{\pi}{3} \), \( t = \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \)

Thus, the integral becomes:
\[ I = \int_{\frac{1}{\sqrt{3}}}^{\sqrt{3}} \frac{1}{2t} \cdot \frac{1}{1 + t^2} \, dt \]

### Step 4: Integration
Now we can integrate:
\[ I = \frac{1}{2} \int_{\frac{1}{\sqrt{3}}}^{\sqrt{3}} \frac{1}{t} \, dt \]
This integral evaluates to:
\[ I = \frac{1}{2} \left[ \ln |t| \right]_{\frac{1}{\sqrt{3}}}^{\sqrt{3}} = \frac{1}{2} \left( \ln(\sqrt{3}) - \ln\left(\frac{1}{\sqrt{3}}\right) \right) \]
Using the property of logarithms \( \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \):
\[ I = \frac{1}{2} \left( \ln(\sqrt{3}) + \ln(\sqrt{3}) \right) = \frac{1}{2} \cdot 2 \ln(\sqrt{3}) = \ln(\sqrt{3}) \]

### Step 5: Final Answer
Thus, the value of the integral is:
\[ I = \ln(\sqrt{3}) = \frac{1}{2} \ln(3) \]