`int_(pi//6)^(pi//4)cosec 2x dx=`

Options

A. `log3`
B. `logsqrt(3)`
C. `log9`
D. `1/2logsqrt(3)`

Answer

To solve the integral \( \int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \csc(2x) \, dx \), we will use a substitution method. Here are the steps to solve the integral:

### Step 1: Substitution
Let \( t = 2x \). Then, the differential \( dx \) can be expressed as:
\[ dx = \frac{dt}{2} \]

### Step 2: Change the limits of integration
When \( x = \frac{\pi}{6} \):
\[ t = 2 \times \frac{\pi}{6} = \frac{\pi}{3} \]
When \( x = \frac{\pi}{4} \):
\[ t = 2 \times \frac{\pi}{4} = \frac{\pi}{2} \]

### Step 3: Rewrite the integral
Now, we can rewrite the integral in terms of \( t \):
\[ \int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \csc(2x) \, dx = \int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \csc(t) \cdot \frac{dt}{2} \]
This simplifies to:
\[ \frac{1}{2} \int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \csc(t) \, dt \]

### Step 4: Integrate \( \csc(t) \)
The integral of \( \csc(t) \) is:
\[ \int \csc(t) \, dt = -\ln |\csc(t) + \cot(t)| + C \]
Thus, we have:
\[ \frac{1}{2} \left[ -\ln |\csc(t) + \cot(t)| \right]_{\frac{\pi}{3}}^{\frac{\pi}{2}} \]

### Step 5: Evaluate the integral at the limits
Now we evaluate the limits:
1. At \( t = \frac{\pi}{2} \):
\[ \csc\left(\frac{\pi}{2}\right) = 1, \quad \cot\left(\frac{\pi}{2}\right) = 0 \quad \Rightarrow \quad \csc\left(\frac{\pi}{2}\right) + \cot\left(\frac{\pi}{2}\right) = 1 \]
Thus, \( -\ln(1) = 0 \).

2. At \( t = \frac{\pi}{3} \):
\[ \csc\left(\frac{\pi}{3}\right) = \frac{2}{\sqrt{3}}, \quad \cot\left(\frac{\pi}{3}\right) = \frac{1}{\sqrt{3}} \quad \Rightarrow \quad \csc\left(\frac{\pi}{3}\right) + \cot\left(\frac{\pi}{3}\right) = \frac{2}{\sqrt{3}} + \frac{1}{\sqrt{3}} = \frac{3}{\sqrt{3}} = \sqrt{3} \]
Thus, \( -\ln\left(\sqrt{3}\right) = -\frac{1}{2} \ln(3) \).

### Step 6: Combine the results
Putting it all together:
\[ \frac{1}{2} \left[ 0 - \left(-\frac{1}{2} \ln(3)\right) \right] = \frac{1}{2} \cdot \frac{1}{2} \ln(3) = \frac{1}{4} \ln(3) \]

### Final Answer
Thus, the value of the integral is:
\[ \int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \csc(2x) \, dx = \frac{1}{4} \ln(3) \]

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