Ex 7.9, 7 - Chapter 7 Class 12 Integrals (Important Question)

Last updated at Dec. 16, 2024 by Teachoo

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Ex 7.9, 7 Evaluate the integrals using substitution ∫_(−1)^(1 )▒〖 𝑑𝑥/(𝑥^2 + 2𝑥 + 5)〗 we can write ∫_(−1)^1▒〖𝑑𝑥/(𝑥^2 + 2𝑥 + 5)=∫_(−1)^1▒𝑑𝑥/((𝑥 + 2𝑥 + 1) + 4)〗 =∫_(−1)^1▒𝑑𝑥/((𝑥 + 1)^2 +〖 2〗^2 ) Putting 𝑥+1=𝑡 Differentiating w.r.t.𝑥 𝑑/𝑑𝑥 (𝑥+1)=𝑑𝑡/𝑑𝑥 1=𝑑𝑡/𝑑𝑥 𝑑𝑥=𝑑𝑡 Hence when 𝑥 varies from – 1 to 1 then 𝑡 varies from 0 to 2 Therefore, ∫_(−1)^1▒〖𝑑𝑥/((𝑥+1)^2 + 2^2 )=∫_0^2▒𝑑𝑡/(𝑡^2 + 2^2 )〗 =[1/2 tan^(−1)⁡〖𝑡/2〗 ]_0^2 =1/2 tan^(−1)⁡〖2/2−1/2 tan^(−1)⁡〖0/2〗〗 =1/2 tan^(−1)⁡〖1−1/2 tan^(−1)⁡0 〗 =1/2 × 𝜋/4−0 =𝝅/𝟖

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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 and Computer Science at Teachoo

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