Find`(d)/(dx)(int_(x^(2))^(x^(3))(1)/(log t)dt)`

Answer

To solve the problem of finding the derivative with respect to \( x \) of the integral from \( x^2 \) to \( x^3 \) of \( \frac{1}{\log t} dt \), we will apply the Leibniz rule for differentiation under the integral sign.

### Step-by-Step Solution:

1. **Identify the Integral**:
We are given the integral:
\[ D = \int_{x^2}^{x^3} \frac{1}{\log t} dt \]

2. **Apply Leibniz Rule**:
According to the Leibniz rule, if we have:
\[ \frac{d}{dx} \left( \int_{a(x)}^{b(x)} f(t) dt \right) = f(b(x)) \cdot b'(x) - f(a(x)) \cdot a'(x) \]
Here, \( a(x) = x^2 \) and \( b(x) = x^3 \), and \( f(t) = \frac{1}{\log t} \).

3. **Calculate Derivatives of Limits**:
- The derivative of the upper limit \( b(x) = x^3 \):
\[ b'(x) = \frac{d}{dx}(x^3) = 3x^2 \]
- The derivative of the lower limit \( a(x) = x^2 \):
\[ a'(x) = \frac{d}{dx}(x^2) = 2x \]

4. **Evaluate the Function at the Limits**:
- Evaluate \( f(b(x)) \):
\[ f(x^3) = \frac{1}{\log(x^3)} = \frac{1}{3 \log x} \]
- Evaluate \( f(a(x)) \):
\[ f(x^2) = \frac{1}{\log(x^2)} = \frac{1}{2 \log x} \]

5. **Substitute into the Leibniz Rule**:
Now substituting into the Leibniz rule:
\[ \frac{dD}{dx} = f(x^3) \cdot b'(x) - f(x^2) \cdot a'(x) \]
This gives:
\[ \frac{dD}{dx} = \left(\frac{1}{3 \log x}\right) \cdot (3x^2) - \left(\frac{1}{2 \log x}\right) \cdot (2x) \]

6. **Simplify the Expression**:
Simplifying the expression:
\[ \frac{dD}{dx} = \frac{3x^2}{3 \log x} - \frac{2x}{2 \log x} \]
This simplifies to:
\[ \frac{dD}{dx} = \frac{x^2}{\log x} - \frac{x}{\log x} \]
Combining the terms:
\[ \frac{dD}{dx} = \frac{x^2 - x}{\log x} \]

### Final Answer:
\[ \frac{d}{dx} \left( \int_{x^2}^{x^3} \frac{1}{\log t} dt \right) = \frac{x(x - 1)}{\log x} \]