If the equation `|sin x |^(2) + |sin x|+b =0` has two distinct roots in `[0, pi][` then the number of integers in the range of b is equal to:

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the equation \( | \sin x |^2 + | \sin x | + b = 0 \) and determine the number of integers in the range of \( b \) such that the equation has two distinct roots in the interval \( [0, \pi] \), we can follow these steps:

### Step 1: Simplify the Equation
Since \( \sin x \) is non-negative in the interval \( [0, \pi] \), we can drop the absolute value signs. Therefore, the equation simplifies to:
\[ \sin^2 x + \sin x + b = 0 \]

### Step 2: Identify the Range of \( \sin x \)
In the interval \( [0, \pi] \), the function \( \sin x \) takes values from \( 0 \) to \( 1 \). Thus, we can denote \( y = \sin x \), where \( y \) ranges from \( 0 \) to \( 1 \).

### Step 3: Rewrite the Equation
Now, we can rewrite the equation in terms of \( y \):
\[ y^2 + y + b = 0 \]

### Step 4: Determine Conditions for Distinct Roots
For the quadratic equation \( y^2 + y + b = 0 \) to have two distinct roots, the discriminant must be positive. The discriminant \( D \) is given by:
\[ D = b^2 - 4ac = 1^2 - 4 \cdot 1 \cdot b = 1 - 4b \]
For the roots to be distinct, we require:
\[ 1 - 4b > 0 \]
This simplifies to:
\[ b < \frac{1}{4} \]

### Step 5: Determine the Range of \( y \)
Next, we need to ensure that the roots \( y_1 \) and \( y_2 \) fall within the range \( [0, 1] \). The sum of the roots \( y_1 + y_2 = -\frac{b}{1} \) must be non-negative:
\[ y_1 + y_2 = -b \geq 0 \implies b \leq 0 \]

### Step 6: Combine the Conditions
From the conditions derived, we have:
1. \( b < \frac{1}{4} \)
2. \( b \leq 0 \)

Thus, the combined condition is:
\[ b \leq 0 \]

### Step 7: Identify the Integer Values of \( b \)
The range of \( b \) that satisfies both conditions is \( (-\infty, 0] \). The integers in this range are \( \ldots, -3, -2, -1, 0 \).

### Step 8: Count the Integer Values
The integers that satisfy this condition are:
- \( 0 \)
- \( -1 \)
- \( -2 \)
- \( -3 \)
- (and so on, infinitely in the negative direction)

However, since we are only interested in the integers up to \( 0 \), we have:
- \( -2, -1, 0 \)

Thus, the total number of integers in the range of \( b \) is \( 3 \).

### Final Answer
The number of integers in the range of \( b \) is \( 3 \).