Example 10 (ii) - Chapter 7 Class 12 Integrals (Term 2)
Last updated at Aug. 20, 2021 by
Transcript
Example 10 Find the following integrals: (ii) β«1β(π₯ + 3)/β(5 β 4π₯ β π₯^2 ) ππ₯ We can write numerator as π₯+3= Aπ/ππ₯ [βπ₯^2β4π₯+5]+ B π₯+3= A[β2π₯β4]+ B π₯+3=β2π΄π₯β4A+B Finding A & B Comparing coefficient of π₯ π₯=β2π΄π₯ 1 =β2A A=(β1)/( 2) Comparing constant term 3=β4A+B 3=β4((β1)/( 2))+B 3=2+B B=3β2=1 Now, we know that π₯+3=A[β2π₯β4]+B π₯+3=(β1)/( 2) [β2π₯β4]+1 Now, our equation is β«1β(π₯+3)/β(5 β 4π₯ + π₯^2 ).ππ₯=β«1β(1/2 [β2π₯ β 4]+1)/β(5 β 4π₯ β π₯^2 ).ππ₯ =β«1β((β1)/2 [β2π₯ β 4]+1)/β(5 β 4π₯ + π₯^2 ) ππ₯+β«1β1/β(5 β 4π₯ β π₯^2 ) ππ₯ =(β1)/2 β«1β(β2π₯ β 4)/β(5 β 4π₯ + π₯^2 ) ππ₯+β«1β1/β(5 β 4π₯ β π₯^2 ) ππ₯ Solving I1 I1 = (β1)/2 β«1βγ(β2π₯ β 4)/β(βπ₯^2 β 4π₯ + 5).ππ₯γ Let t = βπ₯^2 β 4π₯ + 5 Differentiating both sides w.r.t. π₯ β2π₯ β 4 =ππ‘/ππ₯ ππ₯=ππ‘/(β 2π₯ β 4) Now, I1 = (β1)/( 2) β«1βγ(β2π₯ β 4)/β(βπ₯^2 β 4π₯ + 5).ππ₯γ Putting the values of (βπ₯^2β4π₯+5) and ππ₯ I1 = (β1)/( 2) β«1βγ(β2π₯ β 4)/βπ‘.ππ‘/((β2π₯ β4) )γ I1 = (β1)/( 2) β«1βγ1/βπ‘.ππ‘γ I1 = (β1)/( 2) β«1βγ1/(π‘)^(1/2) .ππ‘γ I1 = (β1)/( 2) β«1βγ(π‘)^((β1)/2) ππ‘γ I1 = (β1)/( 2) π‘^((β1)/( 2) + 1)/(((β1)/( 2) + 1) )+πΆ1 I1 = (β1)/( 2) π‘^(1/( 2) )/((1/2) )+πΆ1 I1 = β π‘^(1/( 2) )+πΆ1 I1 = ββπ‘+πΆ1 Putting back π‘=π₯^2β4π₯+5) I1 = ββ(βπ₯^2β4π₯+5)+πΆ1 Solving π°π I2 = β«1βγ1/β(5 β 4π₯ β π₯^2 ).ππ₯γ I2 = β«1βγ1/β(β(π₯^2 + 4π₯ β 5) ) ππ₯γ I2 = β«1βγ1/β(β[π₯^2 + 2(2)(π₯) β 5] ) ππ₯γ I2 = β«1βγ1/β(β[π₯^2 + 2(2)(π₯) + (2)^2 β (2)^2 β 5] ) ππ₯γ I2 = β«1βγ1/β(β[(π₯ + 2)^2 β 4 β 5] ) ππ₯γ I2 = β«1βγ1/β(β[(π₯ + 2)^2 β 9] ) ππ₯γ I2 = β«1βγ1/β(9 β (π₯ + 2)^2 ) ππ₯γ I2 = β«1βγ1/β((3)^2 β (π₯ + 2)^2 ) ππ₯γ It is of form β«1βγ1/β(π^2 β π₯^2 ).ππ₯=γπ ππγ^(β1)β‘γπ₯/π+πΆ2γ γ Replacing π₯ by (π₯+2) and a by 3, we get I2 = sin^(β1)β‘γ((π₯ + 2)/3)+πΆ2γ Now, Putting values of I1 and I2 in (1) β«1βγ(π₯ + 3)/β(5 β 4π₯ β π₯^2 ).ππ₯=(β1)/( 2) β«1β(β 2π₯ β 4)/β(5 β 4π₯ β π₯^2 )γ+β«1β1/β(5 β 4π₯ β π₯^2 ) ππ₯ =ββ(5 β 4π₯ β π₯^2 )+πΆ1+sin^(β1)β‘((π₯ + 2)/3)+πΆ2 =ββ(π β ππ β π^π )+γπππγ^(βπ)β‘γ((π + π)/π)+πͺγ
Examples
Example 1 (i)
Example 1 (ii)
Example 1 (iii)
Example 2 (i)
Example 2 (ii)
Example 2 (iii) Important
Example 3 (i)
Example 3 (ii) Important
Example 3 (iii)
Example 4
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Example 5 (ii)
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Example 5 (iv) Important
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Example 6 (ii) Important
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Example 7 (i)
Example 7 (ii) Important
Example 7 (iii)
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Example 8 (ii) Important
Example 9 (i)
Example 9 (ii) Important
Example 9 (iii) Important
Example 10 (i)
Example 10 (ii) Important You are here
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Example 25 Important Deleted for CBSE Board 2022 Exams
Example 26 Deleted for CBSE Board 2022 Exams
Example 27 (i)
Example 27 (ii) Important
Example 27 (iii)
Example 27 (iv) Important
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Example 25 (Supplementary NCERT) Important Deleted for CBSE Board 2022 Exams
Chapter 7 Class 12 Integrals (Term 2)
Serial order wise
About the Author
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.
