Evaluate : `int " "(x+(1)/(x))^(2) " dx "`

Options

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

Explanation

To evaluate the integral \( \int \left( x + \frac{1}{x} \right)^2 \, dx \), we can follow these steps:

### Step 1: Expand the integrand
We start by expanding the expression \( \left( x + \frac{1}{x} \right)^2 \):

\[ \left( x + \frac{1}{x} \right)^2 = x^2 + 2 \cdot x \cdot \frac{1}{x} + \left( \frac{1}{x} \right)^2 = x^2 + 2 + \frac{1}{x^2} \]

### Step 2: Rewrite the integral
Now we can rewrite the integral as:

\[ \int \left( x^2 + 2 + \frac{1}{x^2} \right) \, dx \]

### Step 3: Separate the integral
We can separate the integral into three parts:

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

### Step 4: Evaluate each integral
Now we evaluate each integral separately:

1. **For \( \int x^2 \, dx \)**:
\[ \int x^2 \, dx = \frac{x^3}{3} \]

2. **For \( \int 2 \, dx \)**:
\[ \int 2 \, dx = 2x \]

3. **For \( \int \frac{1}{x^2} \, dx \)**:
\[ \int \frac{1}{x^2} \, dx = \int x^{-2} \, dx = -\frac{1}{x} \]

### Step 5: Combine the results
Now we combine all the results:

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

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

### Final Answer:
\[ \int \left( x + \frac{1}{x} \right)^2 \, dx = \frac{x^3}{3} + 2x - \frac{1}{x} + C \]

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