If the integral `int(lnx)/(x^(3))dx=(f(x))/(4x^(2))+C`, where `f(e )=-3` and C is the constant of integration, then the value of `f(e^(2))` is equal to

Options

A. 3
B. `-4`
C. `-5`
D. 5

Answer

To solve the integral \( \int \frac{\ln x}{x^3} \, dx = \frac{f(x)}{4x^2} + C \) where \( f(e) = -3 \), we need to find the value of \( f(e^2) \).

### Step-by-Step Solution:

1. **Set Up the Integral**:
We start with the integral:
\[ I = \int \frac{\ln x}{x^3} \, dx \]

2. **Use Integration by Parts**:
We will use integration by parts, where we let:
- \( u = \ln x \) (thus \( du = \frac{1}{x} \, dx \))
- \( dv = \frac{1}{x^3} \, dx \) (thus \( v = -\frac{1}{2x^2} \))

The integration by parts formula is:
\[ I = uv - \int v \, du \]

3. **Apply the Formula**:
Substituting our values into the formula:
\[ I = \ln x \left(-\frac{1}{2x^2}\right) - \int \left(-\frac{1}{2x^2}\right) \left(\frac{1}{x}\right) \, dx \]
This simplifies to:
\[ I = -\frac{\ln x}{2x^2} + \frac{1}{2} \int \frac{1}{x^3} \, dx \]

4. **Integrate the Remaining Integral**:
The remaining integral is:
\[ \int \frac{1}{x^3} \, dx = -\frac{1}{2x^2} \]
Therefore:
\[ I = -\frac{\ln x}{2x^2} - \frac{1}{4x^2} + C \]

5. **Combine Terms**:
We can combine the terms:
\[ I = \frac{-2\ln x - 1}{4x^2} + C \]

6. **Identify \( f(x) \)**:
From the equation \( I = \frac{f(x)}{4x^2} + C \), we can see that:
\[ f(x) = -2\ln x - 1 \]

7. **Find \( f(e) \)**:
We are given \( f(e) = -3 \):
\[ f(e) = -2\ln e - 1 = -2(1) - 1 = -3 \]
This confirms our function \( f(x) \).

8. **Calculate \( f(e^2) \)**:
Now we need to find \( f(e^2) \):
\[ f(e^2) = -2\ln(e^2) - 1 \]
Using the property of logarithms:
\[ \ln(e^2) = 2\ln e = 2 \]
Thus:
\[ f(e^2) = -2(2) - 1 = -4 - 1 = -5 \]

### Final Answer:
The value of \( f(e^2) \) is \( \boxed{-5} \).