Evaluate `int (x^(2)-cos x+(1)/(x))dx`

Options

A. `pi/2 rad`
B. `pi/4 rad`
C. `pi/3 rad`
D. `pi/6 rad`

Explanation

To evaluate the integral \( \int (x^2 - \cos x + \frac{1}{x}) \, dx \), we can break it down into simpler parts.

### Step-by-step Solution:

1. **Break down the integral**:
We can separate the integral into three distinct parts:
\[ \int (x^2 - \cos x + \frac{1}{x}) \, dx = \int x^2 \, dx - \int \cos x \, dx + \int \frac{1}{x} \, dx \]

2. **Integrate \( x^2 \)**:
Using the formula for the integral of \( x^n \):
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
For \( n = 2 \):
\[ \int x^2 \, dx = \frac{x^{2+1}}{2+1} = \frac{x^3}{3} \]

3. **Integrate \( -\cos x \)**:
The integral of \( \cos x \) is:
\[ \int \cos x \, dx = \sin x \]
Therefore,
\[ -\int \cos x \, dx = -\sin x \]

4. **Integrate \( \frac{1}{x} \)**:
The integral of \( \frac{1}{x} \) is:
\[ \int \frac{1}{x} \, dx = \ln |x| \]

5. **Combine the results**:
Now, we can combine all the parts:
\[ \int (x^2 - \cos x + \frac{1}{x}) \, dx = \frac{x^3}{3} - \sin x + \ln |x| + C \]

### Final Answer:
Thus, the final result of the integral is:
\[ \int (x^2 - \cos x + \frac{1}{x}) \, dx = \frac{x^3}{3} - \sin x + \ln |x| + C \]