Integration Full Chapter Explained - https://you.tube/Integration-Class-12
Example 36 - Chapter 7 Class 12 Integrals
Last updated at Dec. 23, 2019 by Teachoo
Transcript
Example 36 Evaluate β«_0^(π/2 )βlogβ‘sinβ‘π₯ ππ₯ Let I1=β«_0^(π/2 )βπππ(π πππ₯) ππ₯ β΄ I1=β«_0^(π/2)βπ ππ(π/2βπ₯)ππ₯ I1= β«_0^(π/2)βπππ(cosβ‘π₯ )ππ₯ Adding (1) and (2) i.e. (1) + (2) I1+ I1=β«_0^(π/2)βγπππ(sinβ‘π₯ )ππ₯+β«_0^(π/2)βπππ(cosβ‘π₯ )ππ₯γ Using Property P4 β«_0^πβγπ(π₯)ππ₯=γ β«_0^πβπ(πβπ₯)ππ₯ β¦(1) β¦(2) 2I1 =β«_0^(π/2)βγlogβ‘[sinβ‘γπ₯ cosβ‘π₯ γ ] ππ₯γ 2I1 = β«_0^(π/2)βγlogβ‘[2sinβ‘γπ₯ cosβ‘π₯ γ/2] ππ₯γ 2I1 = β«_0^(π/2)β[log[2sinβ‘γπ₯ cosβ‘π₯ γ ]βlogβ‘2 ]ππ₯ 2I1 = β«_0^(π/2)β[log[sinβ‘2π₯ ]βlogβ‘2 ]ππ₯ 2I1 = β«_0^(π/2)βlog[sinβ‘2π₯ ]ππ₯ββ«_0^(π/2)βγlog 2 ππ₯γ Solving ππ I2=β«_0^(π/2)βγlog sinβ‘2π₯ ππ₯γ "(Using" logβ‘π + logβ‘π = logβ‘(π.π)) ("Using " log(π/π) = logβ‘(π) β logβ‘(π)) ("Using" sinβ‘2π₯=2 sinβ‘γπ₯ cosβ‘π₯ γ) Let 2π₯=π‘ Differentiating both sides w.r.t.π₯ 2=ππ‘/ππ₯ ππ₯=ππ‘/2 β΄ Putting the values of t and ππ‘ and changing the limits, I2 =β«_0^(π/2)βlog(sinβ‘2π₯ )ππ₯ I2 = β«_0^πβγlog(sinβ‘π‘ ) ππ‘/2γ I2 = 1/2 β«_0^πβlog(sinβ‘π‘ )ππ‘ Using the Property, P6 :- β«_0^2πβπ(π₯)ππ₯=2β«_0^πβπ(π₯)ππ₯, ππ π(2πβπ₯)=π(π₯) Here, π(π‘)=logβ‘π πππ‘ π(2πβπ‘)=π(2πβπ‘)=logβ‘π ππ(2πβπ‘)=logβ‘sinβ‘π‘ Since π(π‘)=π(2πβπ‘) β΄ I2 = 1/2 β«_0^πβlogβ‘sinβ‘γπ‘ ππ‘γ =1/2 Γ2β«_0^(π/2)βlogβ‘sinβ‘γπ‘. ππ‘γ =β«_0^(π/2)βlogβ‘sinβ‘γπ‘. ππ‘γ I2=β«_0^(π/2)βlogβ‘sinβ‘γπ₯ ππ₯γ Now, Using the Property, P0 :- β«_π^πβπ(π₯)ππ₯=β«_π^πβπ(π‘)ππ‘ Putting the value of I2 in equation (3), we get 2I1 =β«_π^(π /π)βπ₯π¨π [πππβ‘ππ ]β‘π π ββ«_0^(π/2)βlog(2)β‘ππ₯ 2I1 = β«_π^(π /π)βπ₯π¨π (πππβ‘π )β‘π π βlog(2) β«_0^(π/2)βγ1.γβ‘ππ₯ 2I1 = ππ β log(2) [π₯]_0^(π/2) 2I1βI1=βlogβ‘2 [π/2β0] I1=βlogβ‘2 [π/2] β΄ ππ=(β π )/π π₯π¨π β‘π
Examples
Example 1
Example 2
Example 3 Important
Example 4
Example 5
Example 6 Important
Example 7 Important
Example 8
Example 9
Example 10 Important
Example 11
Example 12
Example 13 Important
Example 14
Example 15 Important
Example 16 Important
Example 17
Example 18 Important
Example 19
Example 20 Important
Example 21 Important
Example 22 Important
Example 23
Example 24
Example 25 Important Not in Syllabus - CBSE Exams 2021
Example 26 Not in Syllabus - CBSE Exams 2021
Example 27 Important
Example 28
Example 29
Example 30 Important
Example 31
Example 32
Example 33
Example 34 Important
Example 35 Important
Example 36 Important You are here
Example 37
Example 38 Important
Example 39 Important
Example 40 Important
Example 41 Important
Example 42 Important
Example 43 Important
Example 44 Important
Example 25 (Supplementary NCERT) Important Not in Syllabus - CBSE Exams 2021
About the Author
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.