Ex 7.6, 13 - Integrate tan-1 x - Chapter 7 Class 12 - Integration by parts

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Ex 7.6, 13 tan﷮−1﷯ 𝑥 ﷮﷮ tan﷮−1﷯ 𝑥 ﷯ .𝑑𝑥= ﷮﷮ tan﷮−1﷯ 𝑥﷯ 1.𝑑𝑥 ﷯ = tan﷮−1﷯ 𝑥 ﷮﷮1 .﷯ 𝑑𝑥− ﷮﷮ 𝑑 tan﷮−1﷯﷮𝑥﷯﷯﷮𝑑𝑥﷯ ﷮﷮1 .𝑑𝑥﷯﷯﷯𝑑𝑥 = tan﷮−1﷯ 𝑥 𝑥﷯− ﷮﷮ 1﷮1 + 𝑥﷮2﷯﷯﷯ . 𝑥 . 𝑑𝑥 = 𝑥 tan﷮−1﷯ 𝑥− ﷮﷮ 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=𝑥 tan﷮−1﷯ 𝑥− ﷮﷮ 𝑥﷮1 + 𝑥﷮2﷯﷯﷯ . 𝑑𝑥 =𝑥 tan﷮−1﷯ 𝑥− 1﷮2﷯ log ﷮ 1+ 𝑥﷮2﷯﷯﷯+𝐶1﷯ =𝑥 tan﷮−1﷯ 𝑥− 1﷮2﷯ log ﷮ 1+ 𝑥﷮2﷯﷯﷯−𝐶1 =𝒙 𝒕𝒂𝒏﷮−𝟏﷯ 𝒙− 𝟏﷮𝟐﷯ 𝒍𝒐𝒈 ﷮ 𝟏+ 𝒙﷮𝟐﷯﷯﷯+𝑪

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