`int_(0)^(2) |x^(2)+2x-3|dx` is equal to

Options

A. 4
B. 6
C. 3
D. 2

Explanation

To solve the integral \( I = \int_{0}^{2} |x^2 + 2x - 3| \, dx \), we will follow these steps:

### Step 1: Factor the quadratic expression
First, we need to factor the expression inside the absolute value:
\[ x^2 + 2x - 3 = (x - 1)(x + 3) \]
This means the roots of the equation are \( x = 1 \) and \( x = -3 \).

### Step 2: Determine the sign of the expression
Next, we analyze the sign of \( x^2 + 2x - 3 \) in the intervals defined by the roots:
- For \( x < -3 \), both factors are negative, so the product is positive.
- For \( -3 < x < 1 \), \( (x - 1) < 0 \) and \( (x + 3) > 0 \), so the product is negative.
- For \( x > 1 \), both factors are positive, so the product is positive.

Thus, we can summarize the sign of \( x^2 + 2x - 3 \):
- \( x < -3 \): Positive
- \( -3 < x < 1 \): Negative
- \( x > 1 \): Positive

### Step 3: Rewrite the integral with absolute values
Since we are only interested in the interval from \( 0 \) to \( 2 \), we note that:
- From \( 0 \) to \( 1 \), \( x^2 + 2x - 3 < 0 \) (negative).
- From \( 1 \) to \( 2 \), \( x^2 + 2x - 3 > 0 \) (positive).

Thus, we can rewrite the integral:
\[ I = \int_{0}^{1} -(x^2 + 2x - 3) \, dx + \int_{1}^{2} (x^2 + 2x - 3) \, dx \]

### Step 4: Evaluate the integrals
Now, we evaluate each integral separately.

**Integral from 0 to 1:**
\[ \int_{0}^{1} -(x^2 + 2x - 3) \, dx = \int_{0}^{1} (-x^2 - 2x + 3) \, dx \]
Calculating this:
\[ = \left[-\frac{x^3}{3} - x^2 + 3x\right]_{0}^{1} = \left[-\frac{1}{3} - 1 + 3\right] - [0] = -\frac{1}{3} - 1 + 3 = \frac{5}{3} \]

**Integral from 1 to 2:**
\[ \int_{1}^{2} (x^2 + 2x - 3) \, dx \]
Calculating this:
\[ = \left[\frac{x^3}{3} + x^2 - 3x\right]_{1}^{2} = \left[\frac{8}{3} + 4 - 6\right] - \left[\frac{1}{3} + 1 - 3\right] \]
Calculating the limits:
\[ = \left[\frac{8}{3} - 2\right] - \left[\frac{1}{3} - 2\right] = \left[\frac{8}{3} - \frac{6}{3}\right] - \left[\frac{1}{3} - \frac{6}{3}\right] = \frac{2}{3} - \left[-\frac{5}{3}\right] = \frac{2}{3} + \frac{5}{3} = \frac{7}{3} \]

### Step 5: Combine the results
Now, we combine the results from both integrals:
\[ I = \frac{5}{3} + \frac{7}{3} = \frac{12}{3} = 4 \]

### Final Answer
Thus, the value of the integral \( I = \int_{0}^{2} |x^2 + 2x - 3| \, dx \) is \( \boxed{4} \).