`int (x + (1)/(x)) ^(3) dx=`

Options

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

Answer

To solve the integral \( \int \left( x + \frac{1}{x} \right)^3 \, dx \), we will follow these steps:

### Step 1: Expand the integrand
We start by expanding \( \left( x + \frac{1}{x} \right)^3 \) using the binomial theorem:
\[ \left( x + \frac{1}{x} \right)^3 = x^3 + 3x^2 \cdot \frac{1}{x} + 3x \cdot \left(\frac{1}{x}\right)^2 + \left(\frac{1}{x}\right)^3 \]
This simplifies to:
\[ x^3 + 3x + \frac{3}{x} + \frac{1}{x^3} \]

### Step 2: Rewrite the integral
Now we can rewrite the integral:
\[ \int \left( x^3 + 3x + \frac{3}{x} + \frac{1}{x^3} \right) \, dx \]

### Step 3: Separate the integral
We can separate the integral into individual terms:
\[ \int x^3 \, dx + \int 3x \, dx + \int \frac{3}{x} \, dx + \int \frac{1}{x^3} \, dx \]

### Step 4: Integrate each term
Now we will integrate each term separately:
1. For \( \int x^3 \, dx \):
\[ = \frac{x^4}{4} \]
2. For \( \int 3x \, dx \):
\[ = \frac{3x^2}{2} \]
3. For \( \int \frac{3}{x} \, dx \):
\[ = 3 \ln |x| \]
4. For \( \int \frac{1}{x^3} \, dx \):
\[ = -\frac{1}{2x^2} \]

### Step 5: Combine the results
Now we combine all the results:
\[ \int \left( x + \frac{1}{x} \right)^3 \, dx = \frac{x^4}{4} + \frac{3x^2}{2} - \frac{1}{2x^2} + 3 \ln |x| + C \]

### Final Answer
Thus, the final answer is:
\[ \int \left( x + \frac{1}{x} \right)^3 \, dx = \frac{x^4}{4} + \frac{3x^2}{2} - \frac{1}{2x^2} + 3 \ln |x| + C \]

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