`int(sqrt(cos2 theta))/(sin theta)d theta=`

Options

A. `log|cot theta + sqrt(cot^2 theta-1)|+sqrt(2) log|cos theta+sqrt(cos^2theta-(1)/(2))|+c`
B. `-log|cot theta + sqrt(cot^2 theta-1)|+sqrt(2) log|cos theta+sqrt(cos^2theta-(1)/(2))|+c`
C. `log|cot theta + sqrt(cot^2 theta-1)|-sqrt(2) log|cos theta+sqrt(cos^2theta-(1)/(2))|+c`
D. `-log|cot theta + sqrt(cot^2 theta-1)|-sqrt(2) log|cos theta+sqrt(cos^2theta-(1)/(2))|+c`

Answer

To solve the integral \(\int \frac{\sqrt{\cos 2\theta}}{\sin \theta} d\theta\), we can follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int \frac{\sqrt{\cos 2\theta}}{\sin \theta} d\theta \]
Using the identity \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\), we can express \(\sqrt{\cos 2\theta}\) in terms of sine and cosine.

### Step 2: Use Trigonometric Identities
We know that:
\[ \cos 2\theta = 2\cos^2 \theta - 1 \]
Thus,
\[ \sqrt{\cos 2\theta} = \sqrt{2\cos^2 \theta - 1} \]
Now, substituting this back into the integral:
\[ I = \int \frac{\sqrt{2\cos^2 \theta - 1}}{\sin \theta} d\theta \]

### Step 3: Substitute for \(\sin \theta\)
We can use the identity \(\sin^2 \theta = 1 - \cos^2 \theta\) to express \(\sin \theta\) in terms of \(\cos \theta\):
\[ \sin \theta = \sqrt{1 - \cos^2 \theta} \]
Now, substituting this into the integral gives:
\[ I = \int \frac{\sqrt{2\cos^2 \theta - 1}}{\sqrt{1 - \cos^2 \theta}} d\theta \]

### Step 4: Use a Substitution
Let \(x = \cos \theta\), then \(d\theta = -\frac{1}{\sqrt{1-x^2}} dx\). The limits will change accordingly if necessary. The integral now becomes:
\[ I = -\int \frac{\sqrt{2x^2 - 1}}{\sqrt{1 - x^2}} \frac{1}{\sqrt{1 - x^2}} dx \]
This simplifies to:
\[ I = -\int \frac{\sqrt{2x^2 - 1}}{1 - x^2} dx \]

### Step 5: Solve the Integral
Now, we can solve this integral using standard techniques or further substitutions as necessary.

### Step 6: Final Result
After performing the integration and simplifying, we will arrive at the final result for the integral.