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

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Ex7.10, 3 Evaluate the integrals using substitution 0﷮1﷮ sin﷮−1﷯﷮ 2𝑥﷮1 + 𝑥﷮2﷯﷯﷯﷯﷯𝑑𝑥 Let I = 0﷮1﷮ sin﷮−1﷯﷮ 2𝑥﷮1 + 𝑥﷮2﷯﷯﷯﷯﷯𝑑𝑥 Put x = tanϕ Differentiating w.r.t.ϕ 𝑑𝑥﷮𝑑ϕ﷯= 𝑑﷮𝑑ϕ﷯ 𝑡𝑎𝑛ϕ 𝑑𝑥﷮𝑑ϕ﷯= 𝑠𝑒𝑐﷮2﷯ϕ 𝑑𝑥= 𝑠𝑒𝑐﷮2﷯ϕ 𝑑ϕ Hence when x varies from 0 to 1 then 𝜙 varies from 0 to 𝜋﷮4﷯ Therefore we can write integrate I as I = 0﷮ 𝜋﷮4﷯﷮ sin﷮−1﷯﷮ 2𝑡𝑎𝑛𝜙﷮1 + 𝑡𝑎𝑛﷮2﷯𝜙﷯﷯﷯﷯ 𝑠𝑒𝑐﷮2﷯𝜙 𝑑𝜙 I = 0﷮ 𝜋﷮4﷯﷮ sin﷮−1﷯﷮(𝑠𝑖𝑛2𝜙)﷯﷯ 𝑠𝑒𝑐﷮2﷯𝜙 𝑑𝜙 I = 0﷮ 𝜋﷮4﷯﷮ sin﷮−1﷯﷮(𝑠𝑖𝑛2𝜙)﷯﷯ 𝑠𝑒𝑐﷮2﷯𝜙 𝑑𝜙 I = 0﷮ 𝜋﷮4﷯﷮2ϕ 𝑠𝑒𝑐﷮2﷯ϕ 𝑑ϕ ﷯ I = 2 0﷮ 𝜋﷮4﷯﷮ϕ 𝑠𝑒𝑐﷮2﷯ϕ 𝑑ϕ ﷯ I=2× ϕ ∫ 𝑠𝑒𝑐﷮2﷯ϕ𝑑ϕ−∫ 𝑑﷮ ϕ﷯﷯﷮𝑑ϕ﷯∫ 𝑠𝑒𝑐﷮2﷯ϕ𝑑ϕ﷯𝑑ϕ﷯﷮ 𝜋﷮4﷯﷯﷮0﷯ =2 × ϕ tan﷮ϕ﷯− ﷮﷮1﷯× tan﷮ϕ﷯ 𝑑ϕ﷯﷮ 𝜋﷮4﷯﷯﷮0﷯ =2 × ϕ tan﷮ϕ﷯−𝑙𝑜𝑔 sec﷮ϕ﷯﷯﷯﷮ 𝜋﷮4﷯﷯﷮0﷯ =2 𝜋﷮4﷯𝑡𝑎𝑛 𝜋﷮4﷯−𝑙𝑜𝑔 𝑠𝑒𝑐 𝜋﷮4﷯﷯﷯− 0 tan﷮ 0﷯﷯−𝑙𝑜𝑔 sec﷮ 0﷯﷯﷯﷯﷯ =2 𝜋﷮4﷯×1−𝑙𝑜𝑔 ﷮2﷯﷯−0+𝑙𝑜𝑔 1﷯﷯ = 2 𝜋﷮4﷯−𝑙𝑜𝑔 ﷮2﷯−0+0﷯ = 𝜋﷮2﷯−2 𝑙𝑜𝑔 ﷮2﷯ = 𝜋﷮2﷯− 𝑙𝑜𝑔 2﷮ 1﷮2﷯ × 2﷯ = 𝝅﷮𝟐﷯−𝒍𝒐𝒈 𝟐

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