If `f(x)=(x^(2)-1)/x^(3)`, then `intf(x)dx` is

Options

A. `1/x^(2)+1/(2x^(3))+c`
B. `logx+1/(2x^(2))+c`
C. `-1/x^(2)+3/x^(4)+c`
D. `logx+1/(2x^(3))+c`

Answer

To solve the integral of the function \( f(x) = \frac{x^2 - 1}{x^3} \), we will follow these steps:

### Step 1: Rewrite the function
We start by simplifying the function:
\[ f(x) = \frac{x^2 - 1}{x^3} = \frac{x^2}{x^3} - \frac{1}{x^3} = \frac{1}{x} - \frac{1}{x^3} \]

### Step 2: Set up the integral
Now we can set up the integral:
\[ \int f(x) \, dx = \int \left( \frac{1}{x} - \frac{1}{x^3} \right) dx \]

### Step 3: Integrate each term separately
We will integrate each term separately:
1. The integral of \( \frac{1}{x} \) is \( \ln |x| \).
2. The integral of \( -\frac{1}{x^3} \) can be rewritten as \( -x^{-3} \). The integral of \( -x^{-3} \) is:
\[ \int -x^{-3} \, dx = \frac{-x^{-2}}{-2} = \frac{1}{2x^2} \]

### Step 4: Combine the results
Now we combine the results of the integrals:
\[ \int f(x) \, dx = \ln |x| + \frac{1}{2x^2} + C \]
where \( C \) is the constant of integration.

### Final Answer
Thus, the final answer is:
\[ \int f(x) \, dx = \ln |x| + \frac{1}{2x^2} + C \]

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