Ex 7.9, 7 - Chapter 7 Class 12 Integrals

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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