Ex 7.11, 5 - Chapter 7 Class 12 Integrals (Term 2)

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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 Using the Property P2 ∫_𝑎^𝑏▒〖𝑓(𝑥)𝑑𝑥=∫_𝑎^𝑐▒〖𝑓(𝑥)𝑑𝑥+∫_𝑐^𝑏▒𝑓(𝑥)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 =−(((−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