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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‰₯[email protected]βˆ’(π‘₯+2) 𝑖𝑓 π‘₯+2<0)─ ={β–ˆ((π‘₯+2) 𝑖𝑓 π‘₯β‰₯βˆ’,[email protected]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

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.