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 \( \int \frac{\ln x}{x^3} \, dx = \frac{f(x)}{4x^2} + C \) and find the value of \( f(e^2) \), we will follow these steps:

### Step 1: Apply 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 \( v = -\frac{1}{2x^2} \))

Using the integration by parts formula \( \int u \, dv = uv - \int v \, du \):

\[ \int \frac{\ln x}{x^3} \, dx = \ln x \left(-\frac{1}{2x^2}\right) - \int \left(-\frac{1}{2x^2}\right) \left(\frac{1}{x}\right) \, dx \]

### Step 2: Simplify the Integral
This simplifies to:

\[ -\frac{\ln x}{2x^2} + \frac{1}{2} \int \frac{1}{x^3} \, dx \]

Now, we compute the remaining integral:

\[ \int \frac{1}{x^3} \, dx = -\frac{1}{2x^2} \]

Thus,

\[ \frac{1}{2} \int \frac{1}{x^3} \, dx = -\frac{1}{4x^2} \]

### Step 3: Combine Results
Putting it all together, we have:

\[ \int \frac{\ln x}{x^3} \, dx = -\frac{\ln x}{2x^2} - \frac{1}{4x^2} + C \]

### Step 4: Rearrange to Find \( f(x) \)
From the original equation, we have:

\[ -\frac{\ln x}{2x^2} - \frac{1}{4x^2} + C = \frac{f(x)}{4x^2} + C \]

Thus, we can equate:

\[ f(x) = -2\ln x - 1 \]

### Step 5: Evaluate \( f(e) \)
Given \( f(e) = -2\ln e - 1 \):

\[ f(e) = -2(1) - 1 = -3 \]

### Step 6: Evaluate \( f(e^2) \)
Now, we need to find \( f(e^2) \):

\[ f(e^2) = -2\ln(e^2) - 1 = -2(2) - 1 = -4 - 1 = -5 \]

### Final Result
Thus, the value of \( f(e^2) \) is:

\[ \boxed{-5} \]