Find interals of given functions
`x^(2)-2x+1`

Answer

To find the integral of the function \( f(x) = x^2 - 2x + 1 \), we will follow these steps:

### Step 1: Set up the integral
We start by writing the integral of the function:
\[ \int (x^2 - 2x + 1) \, dx \]

### Step 2: Split the integral
We can split the integral into separate terms:
\[ \int (x^2 - 2x + 1) \, dx = \int x^2 \, dx - \int 2x \, dx + \int 1 \, dx \]

### Step 3: Integrate each term
Now, we will integrate each term separately.

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

2. For the second term \( \int 2x \, dx \):
\[ \int 2x \, dx = 2 \cdot \frac{x^{1+1}}{1+1} = 2 \cdot \frac{x^2}{2} = x^2 \]

3. For the third term \( \int 1 \, dx \):
\[ \int 1 \, dx = x \]

### Step 4: Combine the results
Now we combine the results of the integrals:
\[ \int (x^2 - 2x + 1) \, dx = \frac{x^3}{3} - x^2 + x + C \]
where \( C \) is the constant of integration.

### Final Answer
Thus, the integral of the function \( x^2 - 2x + 1 \) is:
\[ \frac{x^3}{3} - x^2 + x + C \]

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