Evaluate the integerals.
`int(x^(3)-2x^(2)+3) dx on R `.

Answer

To evaluate the integral \( \int (x^3 - 2x^2 + 3) \, dx \), we will follow these steps:

### Step 1: Write the integral
We start with the integral:
\[ \int (x^3 - 2x^2 + 3) \, dx \]

### Step 2: Break down the integral
Using the linearity of integration, we can separate the integral into three parts:
\[ \int (x^3 - 2x^2 + 3) \, dx = \int x^3 \, dx - 2 \int x^2 \, dx + \int 3 \, dx \]

### Step 3: Integrate each term
Now we will integrate each term separately using the formula \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \).

1. For \( \int x^3 \, dx \):
\[ \int x^3 \, dx = \frac{x^{3+1}}{3+1} = \frac{x^4}{4} \]

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

3. For \( \int 3 \, dx \):
\[ \int 3 \, dx = 3x \]

### Step 4: Combine the results
Now we combine all the results from the integrations:
\[ \int (x^3 - 2x^2 + 3) \, dx = \frac{x^4}{4} - \frac{2x^3}{3} + 3x + C \]

### Final Answer
Thus, the evaluated integral is:
\[ \int (x^3 - 2x^2 + 3) \, dx = \frac{x^4}{4} - \frac{2x^3}{3} + 3x + C \]