Definite Integration by properties - P4

Chapter 7 Class 12 Integrals
Concept wise

### Transcript

Example 34 Evaluate β«_0^(π/2 )βlogβ‘sinβ‘π₯ ππ₯ Let I1=β«_0^(π/2 )βπππ(π πππ₯) ππ₯ β΄ I1=β«_0^(π/2)βπ ππ(π/2βπ₯)ππ₯ I1= β«_0^(π/2)βπππ(cosβ‘π₯ )ππ₯ Adding (1) and (2) i.e. (1) + (2) I1+ I1=β«_0^(π/2)βγπππ(sinβ‘π₯ )ππ₯+β«_0^(π/2)βπππ(cosβ‘π₯ )ππ₯γ 2I1 =β«_0^(π/2)βγlogβ‘[sinβ‘γπ₯ cosβ‘π₯ γ ] ππ₯γ 2I1 = β«_0^(π/2)βγlogβ‘[2sinβ‘γπ₯ cosβ‘π₯ γ/2] ππ₯γ 2I1 = β«_0^(π/2)β[log[2sinβ‘γπ₯ cosβ‘π₯ γ ]βlogβ‘2 ]ππ₯ 2I1 = β«_0^(π/2)β[log[sinβ‘2π₯ ]βlogβ‘2 ]ππ₯ 2I1 = β«_0^(π/2)βlog[sinβ‘2π₯ ]ππ₯ββ«_0^(π/2)βγlog 2 ππ₯γ Solving ππ I2=β«_0^(π/2)βγlog sinβ‘2π₯ ππ₯γ Let 2π₯=π‘ Differentiating both sides w.r.t.π₯ 2=ππ‘/ππ₯ ππ₯=ππ‘/2 β΄ Putting the values of t and ππ‘ and changing the limits, I2 =β«_0^(π/2)βlog(sinβ‘2π₯ )ππ₯ I2 = β«_0^πβγlog(sinβ‘π‘ ) ππ‘/2γ I2 = 1/2 β«_0^πβlog(sinβ‘π‘ )ππ‘ Here, π(π‘)=logβ‘π πππ‘ π(2πβπ‘)=π(2πβπ‘)=logβ‘π ππ(2πβπ‘)=logβ‘sinβ‘π‘ Since π(π‘)=π(2πβπ‘) β΄ I2 = 1/2 β«_0^πβlogβ‘sinβ‘γπ‘ ππ‘γ =1/2 Γ2β«_0^(π/2)βlogβ‘sinβ‘γπ‘. ππ‘γ =β«_0^(π/2)βlogβ‘sinβ‘γπ‘. ππ‘γ I2=β«_0^(π/2)βlogβ‘sinβ‘γπ₯ ππ₯γ Putting the value of I2 in equation (3), we get 2I1 =β«_π^(π/π)βπ₯π¨π [πππβ‘ππ ]β‘ππ ββ«_0^(π/2)βlog(2)β‘ππ₯ 2I1 = β«_π^(π/π)βπ₯π¨π (πππβ‘π )β‘ππ βlog(2) β«_0^(π/2)βγ1.γβ‘ππ₯ 2I1 = ππ β log(2) [π₯]_0^(π/2) 2I1βI1=βlogβ‘2 [π/2β0] I1=βlogβ‘2 [π/2] β΄ ππ=(β π)/π π₯π¨π β‘π