If `int_(2x^(2))^(x^(3)) (In x)f(t) dt=x^(2)-2x+5`, then f(8)=

Options

A. `(1)/("In"2)`
B. `(1)/(2"In"2)`
C. `(1)/(3"In"2)`
D. `(1)/(4"In"2)`

Explanation

To solve the problem, we need to find the value of \( f(8) \) given that:

\[ \int_{2x^2}^{x^3} \ln x \, f(t) \, dt = x^2 - 2x + 5 \]

### Step 1: Differentiate both sides with respect to \( x \)

Using the Leibniz rule for differentiation under the integral sign, we differentiate the left-hand side:

\[ \frac{d}{dx} \left( \int_{2x^2}^{x^3} \ln x \, f(t) \, dt \right) = \ln x \cdot f(x^3) \cdot \frac{d}{dx}(x^3) - \ln x \cdot f(2x^2) \cdot \frac{d}{dx}(2x^2) \]

Calculating the derivatives:

\[ = \ln x \cdot f(x^3) \cdot 3x^2 - \ln x \cdot f(2x^2) \cdot 4x \]

Now differentiate the right-hand side:

\[ \frac{d}{dx}(x^2 - 2x + 5) = 2x - 2 \]

### Step 2: Set the derivatives equal

Now we have:

\[ \ln x \cdot f(x^3) \cdot 3x^2 - \ln x \cdot f(2x^2) \cdot 4x = 2x - 2 \]

### Step 3: Substitute \( x = 2 \)

We want to find \( f(8) \), and since \( 2^3 = 8 \), we can substitute \( x = 2 \):

\[ \ln(2) \cdot f(8) \cdot 3(2^2) - \ln(2) \cdot f(8) \cdot 4(2) = 2(2) - 2 \]

Calculating the left side:

\[ \ln(2) \cdot f(8) \cdot 12 - \ln(2) \cdot f(8) \cdot 8 = 4 - 2 \]

This simplifies to:

\[ \ln(2) \cdot f(8) \cdot 4 = 2 \]

### Step 4: Solve for \( f(8) \)

Now, we can solve for \( f(8) \):

\[ f(8) = \frac{2}{4 \ln(2)} = \frac{1}{2 \ln(2)} \]

### Final Answer

Thus, the value of \( f(8) \) is:

\[ f(8) = \frac{1}{2 \ln(2)} \]