The number of solutions of the equation `|"sin" (pix)/2+"cos"(pix)/2|=sqrt((In|x|)^(3)+1)` is

Explanation

To solve the equation \( \left| \sin\left(\frac{\pi x}{2}\right) + \cos\left(\frac{\pi x}{2}\right) \right| = \sqrt{(\ln|x|)^3 + 1} \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \left| \sin\left(\frac{\pi x}{2}\right) + \cos\left(\frac{\pi x}{2}\right) \right| = \sqrt{(\ln|x|)^3 + 1} \]

### Step 2: Square both sides
To eliminate the absolute value and the square root, we square both sides:
\[ \left( \sin\left(\frac{\pi x}{2}\right) + \cos\left(\frac{\pi x}{2}\right) \right)^2 = (\ln|x|)^3 + 1 \]

### Step 3: Expand the left side
Using the identity \( \sin^2 A + \cos^2 A = 1 \) and the double angle identity \( 2 \sin A \cos A = \sin(2A) \), we can expand the left side:
\[ \sin^2\left(\frac{\pi x}{2}\right) + \cos^2\left(\frac{\pi x}{2}\right) + 2 \sin\left(\frac{\pi x}{2}\right) \cos\left(\frac{\pi x}{2}\right) = 1 + \sin(\pi x) \]
Thus, we have:
\[ 1 + \sin(\pi x) = (\ln|x|)^3 + 1 \]

### Step 4: Simplify the equation
Subtracting 1 from both sides gives us:
\[ \sin(\pi x) = (\ln|x|)^3 \]

### Step 5: Analyze the functions
Now, we need to analyze the functions \( \sin(\pi x) \) and \( (\ln|x|)^3 \):
- The function \( \sin(\pi x) \) oscillates between -1 and 1 for all real \( x \).
- The function \( (\ln|x|)^3 \) is defined for \( x \neq 0 \) and approaches \( -\infty \) as \( x \) approaches 0 from either side, and it is 0 when \( |x| = 1 \).

### Step 6: Determine intersections
To find the number of solutions, we need to determine how many times these two functions intersect:
- For \( x < 0 \), \( \ln|x| \) is defined and negative, hence \( (\ln|x|)^3 < 0 \).
- For \( 0 < x < 1 \), \( \ln|x| < 0 \) so \( (\ln|x|)^3 < 0 \).
- For \( x = 1 \), \( (\ln|1|)^3 = 0 \).
- For \( x > 1 \), \( \ln|x| > 0 \) and thus \( (\ln|x|)^3 > 0 \).

### Step 7: Count solutions
- From \( x < 0 \), \( \sin(\pi x) \) will intersect \( (\ln|x|)^3 \) at 3 points (as the sine function oscillates).
- From \( 0 < x < 1 \), there will be 3 more intersections.
- From \( x > 1 \), there will be 3 additional intersections.

Thus, the total number of solutions is:
\[ 3 \text{ (for } x < 0\text{)} + 3 \text{ (for } 0 < x < 1\text{)} + 3 \text{ (for } x > 1\text{)} = 9 \]

### Final Answer
The total number of solutions of the equation is **9**.

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