Number of integral solution of the equation ` log_(sin x) sqrt(sin^(2)x)+ log_(cos x)sqrt( cos^(2)x)=`, where ` x in [0,6pi]` is

Answer

To solve the equation

\[ \log_{\sin x} \sqrt{\sin^2 x} + \log_{\cos x} \sqrt{\cos^2 x} = 0 \]

where \( x \in [0, 6\pi] \), we can follow these steps:

### Step 1: Simplify the logarithmic expressions

We know that

\[ \sqrt{\sin^2 x} = |\sin x| \quad \text{and} \quad \sqrt{\cos^2 x} = |\cos x| \]

Thus, we can rewrite the equation as:

\[ \log_{\sin x} |\sin x| + \log_{\cos x} |\cos x| = 0 \]

### Step 2: Use the property of logarithms

Using the property of logarithms, we have:

\[ \log_a b + \log_c d = 0 \implies \log_a b = -\log_c d \]

This means:

\[ \log_{\sin x} |\sin x| = 1 \quad \text{and} \quad \log_{\cos x} |\cos x| = 1 \]

### Step 3: Evaluate the logarithmic expressions

Since \( \log_a a = 1 \), we can conclude:

\[ |\sin x| = \sin x \quad \text{and} \quad |\cos x| = \cos x \]

This holds true when \( \sin x > 0 \) and \( \cos x > 0 \), which occurs in the first quadrant.

### Step 4: Determine the intervals for \( x \)

In the interval \( [0, 6\pi] \), the first quadrant corresponds to:

\[ x \in [0, \frac{\pi}{2}] \quad \text{and repeating every } 2\pi \]

Thus, the intervals where \( \sin x > 0 \) and \( \cos x > 0 \) are:

1. \( [0, \frac{\pi}{2}] \)
2. \( [2\pi, 2\pi + \frac{\pi}{2}] = [2\pi, \frac{5\pi}{2}] \)
3. \( [4\pi, 4\pi + \frac{\pi}{2}] = [4\pi, \frac{9\pi}{2}] \)

### Step 5: Find integral solutions in the specified intervals

Now we need to find the integer values of \( x \) in these intervals:

1. From \( [0, \frac{\pi}{2}] \): The only integer is \( 0 \).
2. From \( [2\pi, \frac{5\pi}{2}] \): The integers are \( 6 \) (since \( 2\pi \approx 6.28 \)).
3. From \( [4\pi, \frac{9\pi}{2}] \): The integers are \( 12 \) (since \( 4\pi \approx 12.56 \)).

### Step 6: Count the integral solutions

The integral solutions we found are:

- \( 0 \) from the first interval
- \( 6 \) from the second interval
- \( 12 \) from the third interval

Thus, the total number of integral solutions in the interval \( [0, 6\pi] \) is:

\[ \text{Total integral solutions} = 3 \]

### Final Answer

The number of integral solutions of the equation is **3**.

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