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Ex 7.11, 18 Class 12 - Evaluate definite integral |x - 1| from 0 to 4

Ex 7.11, 18 - Chapter 7 Class 12 Integrals - Part 2

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Ex 7.11, 18 By using the properties of definite integrals, evaluate the integrals : ∫_0^4β–’|π‘₯βˆ’1| 𝑑π‘₯ |π‘₯βˆ’1|= {β–ˆ( (π‘₯βˆ’1) 𝑖𝑓 π‘₯βˆ’1β‰₯0@βˆ’(π‘₯βˆ’1) 𝑖𝑓 π‘₯βˆ’1<0)─ = {β–ˆ((π‘₯βˆ’1,) 𝑖𝑓 π‘₯β‰₯1@βˆ’(π‘₯βˆ’1) 𝑖𝑓 π‘₯<1)─ ∴ ∫_0^4β–’|π‘₯βˆ’1|𝑑π‘₯=∫_0^1β–’|π‘₯βˆ’1|𝑑π‘₯+∫_1^4β–’|π‘₯βˆ’1|𝑑π‘₯ Using the property, P2 P2 :- ∫_π‘Ž^𝑏▒〖𝑓(π‘₯)𝑑π‘₯=γ€— ∫_π‘Ž^𝑐▒〖𝑓(π‘₯)𝑑π‘₯+∫_𝑐^𝑏▒𝑓(π‘₯)𝑑π‘₯γ€— =∫_0^1β–’γ€–βˆ’(π‘₯βˆ’1)𝑑π‘₯+γ€— ∫_1^4β–’(π‘₯βˆ’1)𝑑π‘₯ =∫_0^1β–’γ€–(βˆ’π‘₯+1)𝑑π‘₯+γ€— ∫_1^4β–’(π‘₯βˆ’1)𝑑π‘₯ =∫_0^1β–’γ€–βˆ’π‘₯ 𝑑π‘₯+γ€— ∫_0^1β–’γ€–1. 𝑑π‘₯+∫_1^4β–’γ€–π‘₯ . 𝑑π‘₯βˆ’βˆ«_1^4β–’γ€–1.𝑑π‘₯γ€—γ€—γ€— =βˆ’[π‘₯^2/2]_0^1+[π‘₯]_0^1βˆ’[π‘₯^2/2]_1^4βˆ’[π‘₯]_1^4 =βˆ’[((1)^2 βˆ’ 0)/2]+[1βˆ’0]+[((4)^2βˆ’(1)^2)/2]βˆ’[4βˆ’1] =βˆ’1/2+1+[(16 βˆ’ 1)/2]βˆ’3 =βˆ’1/2+15/2βˆ’3+1 =(14 )/2βˆ’2= 7 βˆ’ 2 = 5

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