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Ex 7.5
Ex 7.5, 1
Ex 7.5, 2
Ex 7.5, 3 Important
Ex 7.5, 4
Ex 7.5, 5
Ex 7.5, 6 Important You are here
Ex 7.5, 7 Important
Ex 7.5, 8
Ex 7.5, 9 Important
Ex 7.5, 10
Ex 7.5, 11 Important
Ex 7.5, 12
Ex 7.5, 13 Important
Ex 7.5, 14 Important
Ex 7.5, 15
Ex 7.5, 16 Important
Ex 7.5, 17
Ex 7.5, 18 Important
Ex 7.5, 19
Ex 7.5, 20 Important
Ex 7.5, 21 Important
Ex 7.5, 22 (MCQ)
Ex 7.5, 23 (MCQ) Important
Ex 7.6β
Chapter 7 Class 12 Integrals
Serial order wise
- Ex 7.1
- Ex 7.2
- Ex 7.3
- Ex 7.4
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Ex 7.5
- Ex 7.6
- Ex 7.7
- Ex 7.8
- Ex 7.9
- Ex 7.10
- Examples
- Miscellaneous
- Case Based Questions (MCQ)
- NCERT Exemplar MCQ
- Area as a sum
Last updated at April 16, 2024 by Teachoo
Transcript
Ex 7.5, 6Integrate the function (1 β π₯2)/(π₯(1 β 2π₯)) β«1β(1 β π₯2)/(π₯(1 β 2π₯)) ππ₯= β«1β(1 β π₯^2)/(π₯ β 2π₯^2 ) ππ₯ =β«1β(1/2 + (β π₯/2 + 1)/(β2π₯^2+ π₯)) ππ₯ =1/2 β«1βππ₯+1/2 β«1β(βπ₯ + 2)/(β2π₯^2+ π₯) ππ₯ =1/2 β«1βππ₯β1/2 β«1β(π₯ β 2)/π₯(1 β 2π₯) ππ₯ =1/2 β«1βππ₯β1/2 β«1β(π₯ β 2)/π₯(2π₯ β 1) ππ₯ β2π₯^2+π₯ βπ₯^2+π₯/2Now Solving (π₯ β 2)/(π₯ (2π₯ β 1) ) = π΄/π₯ + π΅/(2π₯ β 1) (π₯ β 2)/(π₯ (2π₯ β 1) ) = (π΄(2π₯ β 1) + π΅π₯)/(π₯ (2π₯ β 1) )Cancelling denominator π₯β2=π΄(2π₯β1)+π΅π₯Putting x = 0 in (2) π₯β2=π΄(2π₯β1)+π΅π₯0β2 = π΄(2Γ0β1) + π΅Γ0 β2 = A(β1)β¦(2) β2 = βπ΄ π΄ = 2Similarly Putting x = 1/2 in (2) π₯β2=π΄(2π₯β1)+π΅π₯ 1/2 β 2 = A(1/2Γ2β1)+π΅Γ1/2 (β3)/2 = A(1β1)+π΅Γ1/2 (β3)/2 = AΓ0+ π΅/2 (β3)/2 = π΅/2π΅ = β3Hence we can write it as (π₯ β 2)/(π₯ (2π₯ β 1) ) = π΄/π₯ + π΅/(2π₯ β 1)(π₯ β 2)/(π₯ (2π₯ β 1) ) = 2/π₯ + ((β3))/(2π₯ β 1) = 2/π₯ β 3/(2π₯ β 1)Therefore , from (1) we get, β«1β(1 β π₯^2)/π₯(1 β 2π₯) =1/2 β«1βππ₯+ 1/2 β«1β(2/π₯ β 3/(2π₯ β 1)) ππ₯ =1/2 β«1βππ₯+ 2/2 β«1βγππ₯/π₯ β3/2γ β«1βππ₯/(2π₯ β 1) =1/2 β«1βππ₯+β«1βγππ₯/π₯ + 3/2γ β«1βππ₯/(1 β 2π₯) =1/2 π₯+logβ‘|π₯|+3/2 logβ‘|1 β 2π₯|/(β2) +πΆ =π/π +πππβ‘|π|βπ/π πππβ‘|πβππ|+πͺ
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Made by
Davneet Singh
Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.
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