Check sibling questions

Ex 7.10, 3 - Evaluate integrals sin-1 (2x / 1 + x2) dx - Ex 7.10

Ex 7.10, 3 - Chapter 7 Class 12 Integrals - Part 2
Ex 7.10, 3 - Chapter 7 Class 12 Integrals - Part 3
Ex 7.10, 3 - Chapter 7 Class 12 Integrals - Part 4

Solve all your doubts with Teachoo Black (new monthly pack available now!)


Transcript

Ex 7.10, 3 Evaluate the integrals using substitution ∫_0^1β–’sin^(βˆ’1)⁑(2π‘₯/(1 + π‘₯^2 )) 𝑑π‘₯ Let I = ∫_0^1β–’sin^(βˆ’1)⁑(2π‘₯/(1 + π‘₯^2 )) 𝑑π‘₯ Put x = tan Ο• Differentiating w.r.t.Ο• 𝑑π‘₯/𝑑ϕ=(𝑑 (tan⁑ϕ ))/𝑑ϕ 𝑑π‘₯/𝑑ϕ=〖𝑠𝑒𝑐〗^2 Ο• 𝑑π‘₯=〖𝑠𝑒𝑐〗^2 Ο• 𝑑ϕ Hence when x varies from 0 to 1, πœ™ varies from 0 to πœ‹/4 Therefore we can write integral as I = ∫_0^(πœ‹/4)β–’sin^(βˆ’1)⁑(2π‘‘π‘Žπ‘›πœ™/(1 +γ€–π‘‘π‘Žπ‘›γ€—^2 πœ™)) 〖𝑠𝑒𝑐〗^2 πœ™ π‘‘πœ™ I = ∫_0^(πœ‹/4)β–’sin^(βˆ’1)⁑〖(𝑠𝑖𝑛2πœ™)γ€— 〖𝑠𝑒𝑐〗^2 πœ™ π‘‘πœ™ I = ∫_0^(πœ‹/4)β–’sin^(βˆ’1)⁑〖(𝑠𝑖𝑛2πœ™)γ€— 〖𝑠𝑒𝑐〗^2 πœ™ π‘‘πœ™ I = ∫_0^(πœ‹/4)β–’γ€–2Ο• 〖𝑠𝑒𝑐〗^2 Ο• 𝑑ϕ" " γ€— (Using 2π‘‘π‘Žπ‘›πœ™/(1 +γ€–π‘‘π‘Žπ‘›γ€—^2 πœ™)=sin⁑2πœ™ ) I = 2∫_0^(πœ‹/4)β–’γ€–Ο• 〖𝑠𝑒𝑐〗^2 Ο• 𝑑ϕ" " γ€— I=2Γ—[Ο• βˆ«γ€–π‘ π‘’π‘γ€—^2 Ο• π‘‘Ο•βˆ’βˆ«(𝑑ϕ/π‘‘Ο•βˆ«γ€–π‘ π‘’π‘γ€—^2 Ο• 𝑑ϕ)𝑑ϕ]_0^(πœ‹/4) =2Γ— [Ο• tanβ‘Ο•βˆ’βˆ«1β–’1Γ—tan⁑ϕ 𝑑ϕ]_0^(πœ‹/4) =2Γ— [Ο• tanβ‘Ο•βˆ’π‘™π‘œπ‘”|sec⁑ϕ |]_0^(πœ‹/4) =2[πœ‹/4 π‘‘π‘Žπ‘› πœ‹/4βˆ’π‘™π‘œπ‘”|𝑠𝑒𝑐(πœ‹/4)|βˆ’(0 tan⁑(0)βˆ’π‘™π‘œπ‘”|sec⁑(0) |) ] Using by parts ∫_π‘Ž^𝑏▒〖𝑓(π‘₯)𝑔(π‘₯)𝑑π‘₯=𝑓(π‘₯) ∫_π‘Ž^𝑏▒〖9(π‘₯)βˆ’βˆ«_π‘Ž^𝑏▒〖(𝑓(π‘₯)γ€— γ€— ∫_π‘Ž^𝑏▒〖𝑔(π‘₯)γ€— 𝑑π‘₯) 𝑑π‘₯γ€— Putting 𝑓(π‘₯) = Ο• , g(πœ™)=〖𝑠𝑒𝑐〗^2 Ο• =2 [πœ‹/4Γ—1βˆ’π‘™π‘œπ‘”|√2|βˆ’0+π‘™π‘œπ‘” |1|] = 2 (πœ‹/4βˆ’π‘™π‘œπ‘”βˆš2βˆ’0+0) = πœ‹/2βˆ’2 π‘™π‘œπ‘” √2 = πœ‹/2βˆ’log⁑〖 (√2)^2 γ€— = 𝝅/πŸβˆ’π’π’π’ˆ 𝟐 (∡log⁑1=0) (βˆ΅π‘Ž π‘™π‘œπ‘”π‘₯=π‘™π‘œπ‘”π‘₯^π‘Ž)

Davneet Singh's photo - Co-founder, Teachoo

Made by

Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.