`lim_(x rarr(pi)/(2))((1)/((x-(pi)/(2))^(2))int_(x^(3))^(3)cos(t^((1)/(3)))dt)` is equal to

Options

A. `3(pi/3)^3`
B. `3/2(pi/2)^2`
C. `(pi/2)^3`
D. `(pi/2)^2`

Answer

To solve the limit problem

\[ \lim_{x \to \frac{\pi}{2}} \frac{1}{(x - \frac{\pi}{2})^2} \int_{x^3}^{(\frac{\pi}{2})^3} \cos(t^{\frac{1}{3}}) \, dt, \]

we will use L'Hospital's Rule since we have a \( \frac{0}{0} \) form as \( x \) approaches \( \frac{\pi}{2} \).

### Step-by-Step Solution:

1. **Identify the limit form**: As \( x \to \frac{\pi}{2} \), both the numerator and denominator approach 0. The numerator approaches \( \int_{(\frac{\pi}{2})^3}^{(\frac{\pi}{2})^3} \cos(t^{\frac{1}{3}}) \, dt = 0 \) and the denominator \( (x - \frac{\pi}{2})^2 \to 0 \).

2. **Apply L'Hospital's Rule**: Differentiate the numerator and denominator with respect to \( x \).

- **Numerator**: By the Fundamental Theorem of Calculus, the derivative of the integral is given by:
\[ \frac{d}{dx} \left( \int_{x^3}^{(\frac{\pi}{2})^3} \cos(t^{\frac{1}{3}}) \, dt \right) = -\cos((x^3)^{\frac{1}{3}}) \cdot \frac{d}{dx}(x^3) = -\cos(x) \cdot 3x^2. \]

- **Denominator**: The derivative of \( (x - \frac{\pi}{2})^2 \) is:
\[ 2(x - \frac{\pi}{2}). \]

3. **Rewrite the limit**: Now we have:
\[ \lim_{x \to \frac{\pi}{2}} \frac{-3x^2 \cos(x)}{2(x - \frac{\pi}{2})}. \]

4. **Evaluate the limit again**: As \( x \to \frac{\pi}{2} \), we still have a \( \frac{0}{0} \) form. We apply L'Hospital's Rule again.

- **Differentiate the numerator**:
\[ \frac{d}{dx}(-3x^2 \cos(x)) = -3(2x \cos(x) - x^2 \sin(x)). \]

- **Differentiate the denominator**:
\[ \frac{d}{dx}(2(x - \frac{\pi}{2})) = 2. \]

5. **Rewrite the limit again**:
\[ \lim_{x \to \frac{\pi}{2}} \frac{-3(2x \cos(x) - x^2 \sin(x))}{2}. \]

6. **Substitute \( x = \frac{\pi}{2} \)**:
\[ = \frac{-3}{2} \left( 2 \cdot \frac{\pi}{2} \cdot \cos(\frac{\pi}{2}) - \left(\frac{\pi}{2}\right)^2 \sin(\frac{\pi}{2}) \right). \]
Since \( \cos(\frac{\pi}{2}) = 0 \) and \( \sin(\frac{\pi}{2}) = 1 \), this simplifies to:
\[ = \frac{-3}{2} \left(0 - \frac{\pi^2}{4}\right) = \frac{3\pi^2}{8}. \]

### Final Answer:
Thus, the limit is:
\[ \frac{3\pi^2}{8}. \]