**Problem Statement:**Evaluate the integral:\[\int_{{-3}}^{{5}} |x + 1| \, dx\]**Explanation:**1. **Understand the integrand:** The function inside the integral is \(|x + 1|\), which is the absolute value of \(x + 1\).2. **Find the points where the integrand changes:** The integrand \(|x + 1|\) changes at \(x = -1\) because that's where \(x + 1 = 0\). This divides the interval \([-3, 5]\) into two subintervals: \([-3, -1]\) and \([-1, 5]\).3. **Split the integral:** Break the integral into two parts based on the point where the absolute value changes: \[ \int_{{-3}}^{{5}} |x + 1| \, dx = \int_{{-3}}^{{-1}} |x + 1| \, dx + \int_{{-1}}^{{5}} |x + 1| \, dx \] - In the interval \([-3, -1]\), \(x + 1\) is negative, so \(|x + 1| = -(x + 1) = -x - 1\). - In the interval \([-1, 5]\), \(x + 1\) is non-negative, so \(|x + 1| = x + 1\).4. **Evaluate each integral:** \[ \int_{{-3}}^{{-1}} -x - 1 \, dx + \int_{{-1}}^{{5}} x + 1 \, dx \] The antiderivatives are: \[ -\frac{x^2}{2} - x \bigg|_{{-3}}^{{-1}} + \frac{x^2}{2} + x \bigg|_{{-1}}^{{5}} \]5. **Calculate the definite integrals:** \[ \left( -\frac{(-1)^2}{2} - (-1) \right) - \left( -\frac{(-3)^2}{2}

Name: **Problem Statement:**Evaluate the integral:\[\int_{{-3}}^{{5}} |x +
Uploaded: 2024-03-20T06:46:42.000Z
Channel: Bartleby
Description: **Problem Statement:**Evaluate the integral:\[\int_{{-3}}^{{5}} |x + 1| \, dx\]**Explanation:**1. **Understand the integrand:** The function inside the integral is \(|x + 1|\), which is the absolute value of \(x + 1\).2. **Find the points where the