`(d)/(dx)(int_(x^(2))^((x^(3)) (1)/(logt)dt)` is equal to

Options

A. `x^(2)-x`
B. `(x^(2)-x)logx`
C. `(x^(2)-x)/(logx)`
D. `(x-1^(2))/(logx)`

Explanation

To solve the problem \(\frac{d}{dx}\left(\int_{x^2}^{x^3} \frac{1}{\log t} dt\right)\), we will use the Leibniz rule for differentiation under the integral sign.

### Step-by-Step Solution:

1. **Identify the Integral and Its Limits**:
The integral we are dealing with is:
\[ F(x) = \int_{x^2}^{x^3} \frac{1}{\log t} dt \]

2. **Apply Leibniz Rule**:
According to Leibniz's rule, if \( F(x) = \int_{g(x)}^{h(x)} f(t) dt \), then:
\[ \frac{d}{dx} F(x) = f(h(x)) \cdot h'(x) - f(g(x)) \cdot g'(x) \]
Here, \( f(t) = \frac{1}{\log t} \), \( g(x) = x^2 \), and \( h(x) = x^3 \).

3. **Calculate the Derivatives of the Limits**:
We need to find \( h'(x) \) and \( g'(x) \):
\[ h'(x) = \frac{d}{dx}(x^3) = 3x^2 \]
\[ g'(x) = \frac{d}{dx}(x^2) = 2x \]

4. **Evaluate the Function at the Limits**:
Now, we need to evaluate \( f(h(x)) \) and \( f(g(x)) \):
\[ f(h(x)) = f(x^3) = \frac{1}{\log(x^3)} = \frac{1}{3 \log x} \quad \text{(using the property of logarithms)} \]
\[ f(g(x)) = f(x^2) = \frac{1}{\log(x^2)} = \frac{1}{2 \log x} \quad \text{(using the property of logarithms)} \]

5. **Substitute into the Leibniz Rule**:
Now substitute everything into the Leibniz rule:
\[ \frac{d}{dx} F(x) = \frac{1}{3 \log x} \cdot 3x^2 - \frac{1}{2 \log x} \cdot 2x \]

6. **Simplify the Expression**:
Simplifying the expression gives:
\[ \frac{d}{dx} F(x) = \frac{3x^2}{3 \log x} - \frac{2x}{2 \log x} = \frac{x^2}{\log x} - \frac{x}{\log x} \]
\[ = \frac{x^2 - x}{\log x} \]

7. **Final Result**:
Thus, the final result is:
\[ \frac{d}{dx}\left(\int_{x^2}^{x^3} \frac{1}{\log t} dt\right) = \frac{x^2 - x}{\log x} \]