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Ex 7.6
Ex 7.6, 1
Ex 7.6, 2 Important
Ex 7.6, 3
Ex 7.6, 4
Ex 7.6, 5 Important
Ex 7.6, 6
Ex 7.6, 7 Important
Ex 7.6, 8
Ex 7.6, 9
Ex 7.6, 10 Important
Ex 7.6, 11
Ex 7.6, 12
Ex 7.6, 13 Important
Ex 7.6, 14 Important
Ex 7.6, 15
Ex 7.6, 16
Ex 7.6, 17
Ex 7.6, 18 Important
Ex 7.6, 19
Ex 7.6, 20 Important
Ex 7.6, 21
Ex 7.6, 22 Important You are here
Ex 7.6, 23 (MCQ)
Ex 7.6, 24 (MCQ) Important
Ex 7.7β
Chapter 7 Class 12 Integrals
Serial order wise
- Ex 7.1
- Ex 7.2
- Ex 7.3
- Ex 7.4
- Ex 7.5
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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 7.6, 22Integrate the function sin^(β1) (2π₯/(1 + π₯2))Simplifying the given function sin^(β1) (2π₯/(1 + π₯2))Let π₯=tanβ‘π‘ β΄ π‘=tan^(β1)β‘(π₯)β΄ sin^(β1) (2π₯/(1 + π₯2))=sin^(β1) ((2 tanβ‘π‘" " )/(1 + tan^2β‘π‘ )) =sin^(β1) (sinβ‘2π‘ ) = 2t Ex 7.6, 22Integrate the function sin^(β1) (2π₯/(1 + π₯2))Simplifying the given function sin^(β1) (2π₯/(1 + π₯2))Let π₯=tanβ‘π‘ β΄ π‘=tan^(β1)β‘(π₯)β΄ sin^(β1) (2π₯/(1 + π₯2))=sin^(β1) ((2 tanβ‘π‘" " )/(1 + tan^2β‘π‘ )) =sin^(β1) (sinβ‘2π‘ ) = 2t (ππ πππ sinβ‘2π=(2 tanβ‘π" " )/(1 + tan^2β‘π )) (ππ πππ sin^(β1) (sinβ‘π₯ )=π₯) =2 tan^(β1)β‘π₯ Thus, our function becomesβ«1βγsin^(β1) (2π₯/(1 + π₯2)) ππ₯γ = 2β«1βγtan^(β1)β‘π₯ ππ₯γ =2β«1βγ(tan^(β1) π₯) 1.ππ₯ " " γ = 2 tan^(β1) π₯β«1βγ1 .γ ππ₯β2β«1β(π(tan^(β1)β‘π₯ )/ππ₯ β«1βγ1 .ππ₯γ) ππ₯ = 2tan^(β1) π₯ (π₯)β2β«1β1/(1 + π₯^2 ) . π₯ . ππ₯ Now we know that β«1βγπ(π₯) πβ‘(π₯) γ ππ₯=π(π₯) β«1βπ(π₯) ππ₯ββ«1β(πβ²(π₯)β«1βπ(π₯) ππ₯) ππ₯Putting f(x) = tanβ1 x and g(x) = 1(ππ πππ π‘=tan^(β1)β‘(π₯) ) = 2π₯ tan^(β1) π₯β2β«1βπ₯/(1 + π₯^2 ) . ππ₯ Solving I1I1 = β«1βπ₯/(1 + π₯^2 ) . ππ₯" "Let 1 + π₯^2=π‘Differentiating both sides π€.π.π‘.π₯ 0 + 2π₯=ππ‘/ππ₯ ππ₯=ππ‘/2π₯Our equation becomesI1 = β«1βπ₯/(1 + π₯^2 ) . ππ₯" " Putting the value of (1+π₯^2 ) = t and ππ₯ = ππ‘/( 2π₯) , we getI1 = β«1βπ₯/π‘ . ππ‘/2π₯ I1 = 1/2 β«1β1/π‘ . ππ‘ I1 = 1/2 logβ‘γ |π‘|γ+πΆ1 I1 = 1/2 logβ‘γ |1+π₯^2 |γ+πΆ1Putting the value of I1 in (1) , β«1βγsin^(β1) (2π₯/(1 + π₯2)) γ .ππ₯=2β«1βγ" " tan^(β1) π₯" " γ .ππ₯=2π₯ tan^(β1) π₯β2β«1βπ₯/(1 + π₯^2 ) . ππ₯ =2π₯ tan^(β1) π₯β2(1/2 γlog γβ‘|1+π₯^2 |+πΆ1)(As t = 1 + x2) =2π₯ tan^(β1) π₯βγlog γβ‘|1+π₯^2 |β2πΆ1 =ππ γπππγ^(βπ) πβγπππ γβ‘(π+π^π )+πͺ As 1 + x2 is always positive|1+π₯^2 | = 1 + x2
Next: Ex 7.6, 23 (MCQ) β Go Ad-free
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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.