**Given: \( f(x) \) is an even function (this is important!),**\[ \int_{-3}^{0} f(x) \, dx = -4, \quad \int_{0}^{5} f(x) \, dx = 6, \quad \int_{3}^{8} f(x) \, dx = 5 \]**a.** \[ \int_{5}^{0} 4f(x) \, dx = \]**b.** \[ \int_{3}^{5} f(x) \, dx = \]**c.** \[ \int_{2}^{7} \left( f(x+1) + 4 \right) \, dx = \]- **Note on Even Function:** An even function is symmetric about the y-axis, meaning \( f(x) = f(-x) \).- **Explanation:** - The problem provides three definite integrals of an even function and asks to find values of other integrals based on given information. - For each section a, b, and c, students are tasked with calculating new integrals using properties of definite integrals and the given information about \( f(x) \). - In section a, note that reversing the limits of integration changes the sign of the integral. - In section b, consider properties of definite integrals and the intervals provided. - In section c, apply the integration of constants and functions with shifted arguments.

Name: **Given: \( f(x) \) is an even function (this is important!),**\[ \in
Uploaded: 2024-03-20T06:46:42.000Z
Channel: Bartleby
Description: **Given: \( f(x) \) is an even function (this is important!),**\[ \int_{-3}^{0} f(x) \, dx = -4, \quad \int_{0}^{5} f(x) \, dx = 6, \quad \int_{3}^{8} f(x) \, dx = 5 \]**a.** \[ \int_{5}^{0} 4f(x) \, dx = \]**b.** \[ \int_{3}^{5} f(x) \, dx = \]**c.

Given fx is an even function ∫-30fxdx=-4∫05fxdx=6∫38fxdx=5a.) ∫504fxdx Since, ∫abc×fxdx=-c∫bafxdx ∫504fxdx=-4∫05fxdx=-46=-24b.) ∫35fxdx Since, fx is an even function ∫0afxdx=∫-a0fxdx∫03fxdx=∫-30fxdx∫03fxdx=-4 Hence, ∫05fxdx=∫03fxdx+∫35fxdx∫35fxdx=∫05fxdx-∫03fxdx∫35fxdx=6--4∫35fxdx=10c.) ∫27fx+1+4dx Using substitution put x+1=tdx=dtx=2⇒t=3x=7⇒t=8∫27fx+1+4dx=∫38ft+4dt∫27fx+1+4dx=∫38ftdt + ∫384dt∫27fx+1+4dx=5+4t38∫27fx+1+4dx=5+48-3∫27fx+1+4dx=5+20∫27fx+1+4dx=25