The minimum integral value of  3\left( {\frac{{1 - {{\sin }^3}x}}{{{{\cos }^2}x}}} \right)  is

We need to determine the minimum integral value of the expression:

\[
y = 3\left( \frac{1 - \sin^3 x}{\cos^2 x} \right).
\]

### Step 1: Rewrite the expression using trigonometric identities.

First, recall that:

\[
\cos^2 x = 1 - \sin^2 x.
\]

So, the expression becomes:

\[
y = 3 \cdot \frac{1 - \sin^3 x}{1 - \sin^2 x}.
\]

Let \( t = \sin x \). Then \( t \in [-1, 1] \), and the expression can be rewritten as:

\[
y = 3 \cdot \frac{1 - t^3}{1 - t^2}.
\]

### Step 2: Simplify the expression.

We need to simplify the expression:

\[
y = 3 \cdot \frac{1 - t^3}{1 - t^2}.
\]

The numerator \( 1 - t^3 \) can be factored as:

\[
1 - t^3 = (1 - t)(1 + t + t^2).
\]

So, the expression for \( y \) becomes:

\[
y = 3 \cdot \frac{(1 - t)(1 + t + t^2)}{1 - t^2}.
\]

The denominator \( 1 - t^2 \) can be factored as:

\[
1 - t^2 = (1 - t)(1 + t).
\]

Thus, the expression simplifies to:

\[
y = 3 \cdot \frac{1 + t + t^2}{1 + t}.
\]

### Step 3: Analyze the expression.

We need to find the minimum value of:

\[
y = 3 \cdot \frac{1 + t + t^2}{1 + t} \quad \text{for} \quad t \in [-1, 1].
\]

### Step 4: Investigate the critical points and boundary values.

1. **At \( t = -1 \):**
   \[
   y = 3 \cdot \frac{1 - 1 + 1}{1 - 1} \quad (\text{undefined}).
   \]
   So, \( y \) is undefined at \( t = -1 \).

2. **At \( t = 0 \):**
   \[
   y = 3 \cdot \frac{1 + 0 + 0^2}{1 + 0} = 3 \cdot \frac{1}{1} = 3.
   \]

3. **At \( t = 1 \):**
   \[
   y = 3 \cdot \frac{1 + 1 + 1^2}{1 + 1} = 3 \cdot \frac{1 + 1 + 1}{2} = 3 \cdot \frac{3}{2} = \frac{9}{2} = 4.5.
   \]

### Step 5: Determine the minimum value.

The minimum value of \( y \) occurs at \( t = 0 \), where \( y = 3 \). Since we are asked for the minimum integral value, the answer is:

\[
\boxed{3}.
\]

Would you like more details or have any other questions?

Here are 5 related questions:
1. How do you find the minimum value of a function over an interval?
2. What is the process of simplifying trigonometric expressions?
3. Can trigonometric identities be applied to non-standard angles?
4. How do you analyze critical points of functions using calculus?
5. What role does factoring play in simplifying polynomial expressions?

**Tip:** Always check the boundary points when finding the minimum or maximum of a function over a closed interval.

Explore how to find the minimum integral value of the expression 3(1 - sin^3(x) / cos^2(x)) using trigonometric identities and optimization techniques. Detailed step-by-step solution included.

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