Ex 7.11, 5 - Using properties of definite integrals, |x + 2| dx

Ex 7.11, 5 By using the properties of definite integrals, evaluate the integrals : ∫_(−5)^5▒〖 |𝑥+2| 〗 𝑑𝑥 |𝑥+2|={█((𝑥+2) 𝑖𝑓 𝑥+2≥0@−(𝑥+2) 𝑖𝑓 𝑥+2<0)┤ ={█((𝑥+2) 𝑖𝑓 𝑥≥−,2@−(𝑥+2) 𝑖𝑓 𝑥<−2)┤ ∴ ∫_(−5)^5▒〖|𝑥+2|𝑑𝑥=∫_(−5)^(−2)▒〖|𝑥+2|𝑑𝑥+〗〗 ∫_(−2)^5▒|𝑥+2|𝑑𝑥 =∫_(−5)^(−2)▒〖(𝑥+2)𝑑𝑥+〗 ∫_(−2)^5▒〖+(𝑥+2)𝑑𝑥〗 =∫_(−5)^(−2)▒〖𝑥𝑑𝑥−〗 ∫_(−5)^(−2)▒〖2𝑑𝑥−∫_(−2)^5▒〖𝑥𝑑𝑥−∫_(−2)^5▒2𝑑𝑥〗〗 =[𝑥^2/2]_(−5)^(−2)+2[𝑥]_(−5)^(−2)−[𝑥^2/2]_(−2)^5−2[𝑥]_(−2)^5 =[𝑥^2/2]_(−5)^(−2)+2[𝑥]_(−5)^(−2)+[𝑥^2/2]_(−2)^5−2[𝑥]_(−2)^5 =(−(−2)^2+ (−5)^2)/2−2[−2−(−5)]+[((5)^2−(−2)^2)/2+2[5−(−2)]] =(4 + 25)/2−2[−2+5]+[(25 − 4)/2]+2[5+2] =(+ 21)/2−2[3]+21/2+2[7] =(+ 21)/2+(− 21)/2−6+14 =42/2+8 = 21+8 = 𝟐𝟗