Ex 7.6, 22 - Integrate sin-1 (2x / 1 + x2) - Class 12 CBSE - Ex 7.6

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Ex 7.6, 22 sin﷮−1﷯ 2𝑥﷮1 + 𝑥2﷯﷯ Simplifying the given function sin﷮−1﷯ 2𝑥﷮1 + 𝑥2﷯﷯ Put 𝑥=tan⁡𝑡 ∴ 𝑡= tan﷮−1﷯﷮ 𝑥﷯﷯ ∴ sin﷮−1﷯ 2𝑥﷮1 + 𝑥2﷯﷯= sin﷮−1﷯ 2 tan⁡𝑡 ﷮1 + tan﷮2﷯﷮𝑡﷯﷯﷯ = sin﷮−1﷯ sin﷮2𝑡﷯﷯ = 2t =2 tan﷮−1﷯﷮𝑥﷯ Thus, our function becomes ﷮﷮ sin﷮−1﷯ 2𝑥﷮1 + 𝑥2﷯﷯ 𝑑𝑥﷯ = 2 ﷮﷮ tan﷮−1﷯﷮𝑥﷯ 𝑑𝑥﷯ =2 ﷮﷮ tan﷮−1﷯ 𝑥﷯ 1.𝑑𝑥 ﷯ = 2 tan﷮−1﷯ 𝑥 ﷮﷮1 .﷯ 𝑑𝑥−2 ﷮﷮ 𝑑 tan﷮−1﷯﷮𝑥﷯﷯﷮𝑑𝑥﷯ ﷮﷮1 .𝑑𝑥﷯﷯﷯𝑑𝑥 = 2 tan﷮−1﷯ 𝑥 𝑥﷯−2 ﷮﷮ 1﷮1 + 𝑥﷮2﷯﷯﷯ . 𝑥 . 𝑑𝑥 = 2𝑥 tan﷮−1﷯ 𝑥−2 ﷮﷮ 𝑥﷮1 + 𝑥﷮2﷯﷯﷯ . 𝑑𝑥 Solving I1 I1 = ﷮﷮ 𝑥﷮1 + 𝑥﷮2﷯﷯﷯ . 𝑑𝑥 Let 1 + 𝑥﷮2﷯=𝑡 Differentiating both sides 𝑤.𝑟.𝑡.𝑥 0 + 2𝑥= 𝑑𝑡﷮𝑑𝑥﷯ 𝑑𝑥= 𝑑𝑡﷮2𝑥﷯ Now, I1 = ﷮﷮ 𝑥﷮1 + 𝑥﷮2﷯﷯﷯ . 𝑑𝑥 Putting the value of 1+ 𝑥﷮2﷯﷯ = t and 𝑑𝑥 = 𝑑𝑡﷮ 2𝑥﷯ , we get I1 = ﷮﷮ 𝑥﷮𝑡﷯﷯ . 𝑑𝑡﷮2𝑥﷯ I1 = 1﷮2﷯ ﷮﷮ 1﷮𝑡﷯﷯ . 𝑑𝑡 I1 = 1﷮2﷯ log﷮ 𝑡﷯﷯+𝐶1 I1 = 1﷮2﷯ log﷮ 1+ 𝑥﷮2﷯﷯﷯+𝐶1 Putting the value of I1 in (1) , ﷮﷮ tan﷮−1﷯ 𝑥 ﷯ .𝑑𝑥=2𝑥 tan﷮−1﷯ 𝑥−2 ﷮﷮ 𝑥﷮1 + 𝑥﷮2﷯﷯﷯ . 𝑑𝑥 =2𝑥 tan﷮−1﷯ 𝑥−2 1﷮2﷯ log ﷮ 1+ 𝑥﷮2﷯﷯﷯+𝐶1﷯ =2𝑥 tan﷮−1﷯ 𝑥− log ﷮ 1+ 𝑥﷮2﷯﷯﷯−2𝐶1 =𝟐𝒙 𝒕𝒂𝒏﷮−𝟏﷯ 𝒙− 𝒍𝒐𝒈 ﷮ 𝟏+ 𝒙﷮𝟐﷯﷯﷯+𝑪

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