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

Ex 7.11, 5 - Chapter 7 Class 12 Integrals - Part 2
Ex 7.11, 5 - Chapter 7 Class 12 Integrals - Part 3


Transcript

Ex 7.11, 5 By using the properties of definite integrals, evaluate the integrals : ∫_(βˆ’5)^5β–’γ€– |π‘₯+2| γ€— 𝑑π‘₯ |π‘₯+2|={β–ˆ((π‘₯+2) 𝑖𝑓 π‘₯+2β‰₯0@βˆ’(π‘₯+2) 𝑖𝑓 π‘₯+2<0)─ ={β–ˆ((π‘₯+2) 𝑖𝑓 π‘₯β‰₯βˆ’,2@βˆ’(π‘₯+2) 𝑖𝑓 π‘₯<βˆ’2)─ ∴ ∫_(βˆ’5)^5β–’γ€–|π‘₯+2|𝑑π‘₯=∫_(βˆ’5)^(βˆ’2)β–’γ€–|π‘₯+2|𝑑π‘₯+γ€—γ€— ∫_(βˆ’2)^5β–’|π‘₯+2|𝑑π‘₯ Using the Property P2 ∫_π‘Ž^𝑏▒〖𝑓(π‘₯)𝑑π‘₯=∫_π‘Ž^𝑐▒〖𝑓(π‘₯)𝑑π‘₯+∫_𝑐^𝑏▒𝑓(π‘₯)𝑑π‘₯γ€—γ€— =∫_(βˆ’5)^(βˆ’2)β–’γ€–βˆ’(π‘₯+2)𝑑π‘₯+γ€— ∫_(βˆ’2)^5β–’(π‘₯+2)𝑑π‘₯ =βˆ’βˆ«_(βˆ’5)^(βˆ’2)β–’γ€–π‘₯𝑑π‘₯βˆ’γ€— ∫_(βˆ’5)^(βˆ’2)β–’2𝑑π‘₯+∫_(βˆ’2)^5β–’π‘₯𝑑π‘₯+∫_(βˆ’2)^5β–’2𝑑π‘₯ =βˆ’[π‘₯^2/2]_(βˆ’5)^(βˆ’2)βˆ’2[π‘₯]_(βˆ’5)^(βˆ’2)+[π‘₯^2/2]_(βˆ’2)^5+2[π‘₯]_(βˆ’2)^5 =βˆ’(((βˆ’2)^2 βˆ’ (βˆ’5)^2)/2)βˆ’2[βˆ’2βˆ’(βˆ’5)]+[((5)^2 βˆ’ (βˆ’2)^2)/2] +2 [5βˆ’(βˆ’2)] =βˆ’((4 βˆ’ 25)/2)βˆ’2[βˆ’2+5]+[(25 βˆ’ 4)/2]+2[5+2] =βˆ’((βˆ’21)/2)βˆ’2[3]+21/2+2[7] =21/2+21/2βˆ’6+14 =42/2+8 = 21+8 = πŸπŸ—

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.