`int(x+(1)/(x))^(2)dx=`

Options

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

Answer

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

### Step 1: Expand the integrand
First, we need to expand 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 using the expanded form:

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

### Step 3: Split the integral
We can split the integral into three separate integrals:

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

### Step 4: Integrate each term
Now we will integrate each term 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 from the integrations:

\[ \frac{x^3}{3} + 2x - \frac{1}{x} + C \]

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

### Final Answer
Thus, the final result of the integral is:

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

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