If `f'(x)= x-(3)/(x^(3)), f(1) = (11)/(2)` find f(x)

Answer

To find the function \( f(x) \) given its derivative \( f'(x) = x - \frac{3}{x^3} \) and the initial condition \( f(1) = \frac{11}{2} \), we will follow these steps:

### Step 1: Integrate \( f'(x) \)

We start by integrating \( f'(x) \) to find \( f(x) \):

\[ f(x) = \int f'(x) \, dx = \int \left( x - \frac{3}{x^3} \right) dx \]

### Step 2: Break down the integral

We can split the integral into two parts:

\[ f(x) = \int x \, dx - \int \frac{3}{x^3} \, dx \]

### Step 3: Compute the first integral

The integral of \( x \) is:

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

### Step 4: Compute the second integral

The integral of \( \frac{3}{x^3} \) can be rewritten as \( 3x^{-3} \):

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

### Step 5: Combine the results

Now, combining both integrals, we have:

\[ f(x) = \frac{x^2}{2} + \frac{3}{2x^2} + C \]

where \( C \) is the constant of integration.

### Step 6: Use the initial condition to find \( C \)

We know that \( f(1) = \frac{11}{2} \). Substituting \( x = 1 \):

\[ f(1) = \frac{1^2}{2} + \frac{3}{2 \cdot 1^2} + C = \frac{1}{2} + \frac{3}{2} + C = \frac{4}{2} + C = 2 + C \]

Setting this equal to \( \frac{11}{2} \):

\[ 2 + C = \frac{11}{2} \]

### Step 7: Solve for \( C \)

Subtracting 2 from both sides:

\[ C = \frac{11}{2} - 2 = \frac{11}{2} - \frac{4}{2} = \frac{7}{2} \]

### Step 8: Write the final function

Now we can write the final expression for \( f(x) \):

\[ f(x) = \frac{x^2}{2} + \frac{3}{2x^2} + \frac{7}{2} \]

### Summary of the solution

Thus, the function \( f(x) \) is:

\[ f(x) = \frac{x^2}{2} + \frac{3}{2x^2} + \frac{7}{2} \]