If `f(x)=int_(x)^(x^(2))(dt)/((logt)^(2)),xne0` then `f(x)` is

Options

A. monotonically increasing in `(2,oo)`
B. monotonically decreasing in (1, 2)
C. monotonically decreasing in `(2,oo)`
D. monotonically decreasing in (0, 1)

Answer

To solve the problem where \( f(x) = \int_{x}^{x^2} \frac{dt}{(\log t)^2} \) for \( x \neq 0 \), we will use Leibniz's rule for differentiation under the integral sign.

### Step-by-Step Solution:

**Step 1: Apply Leibniz's Rule**
Leibniz's rule states that if \( F(x) = \int_{a(x)}^{b(x)} f(t) dt \), then the derivative \( F'(x) \) is given by:
\[ F'(x) = f(b(x)) \cdot b'(x) - f(a(x)) \cdot a'(x) \]
In our case:
- \( a(x) = x \)
- \( b(x) = x^2 \)
- \( f(t) = \frac{1}{(\log t)^2} \)

**Step 2: Calculate the Derivatives of the Limits**
We need to find the derivatives of the limits:
- \( a'(x) = 1 \)
- \( b'(x) = 2x \)

**Step 3: Evaluate the Function at the Limits**
Now we evaluate \( f(t) \) at the limits:
- At the upper limit \( b(x) = x^2 \):
\[ f(b(x)) = f(x^2) = \frac{1}{(\log(x^2))^2} = \frac{1}{(2 \log x)^2} = \frac{1}{4 (\log x)^2} \]
- At the lower limit \( a(x) = x \):
\[ f(a(x)) = f(x) = \frac{1}{(\log x)^2} \]

**Step 4: Substitute into Leibniz's Rule**
Now substituting into Leibniz's rule:
\[ f'(x) = f(x^2) \cdot b'(x) - f(x) \cdot a'(x) \]
\[ f'(x) = \frac{1}{4 (\log x)^2} \cdot (2x) - \frac{1}{(\log x)^2} \cdot (1) \]
\[ f'(x) = \frac{2x}{4 (\log x)^2} - \frac{1}{(\log x)^2} \]
\[ f'(x) = \frac{x}{2 (\log x)^2} - \frac{1}{(\log x)^2} \]

**Step 5: Combine the Terms**
Combine the terms over a common denominator:
\[ f'(x) = \frac{x - 2}{2 (\log x)^2} \]

**Step 6: Analyze the Sign of \( f'(x) \)**
To determine where \( f(x) \) is increasing or decreasing, we analyze the sign of \( f'(x) \):
- \( f'(x) > 0 \) when \( x - 2 > 0 \) or \( x > 2 \) (function is increasing).
- \( f'(x) < 0 \) when \( x - 2 < 0 \) or \( x < 2 \) (function is decreasing).

### Conclusion
Thus, \( f(x) \) is:
- Monotonically increasing for \( x > 2 \)
- Monotonically decreasing for \( 0 < x < 2 \)