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Ex 7.6, 14 - Integrate x (log x)2 - Chapter 7 Class 12 - Ex 7.6

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Ex 7.6, 14 𝑥(log⁡𝑥)﷮2﷯ ﷮﷮𝑥 log﷮𝑥﷯﷯﷮2﷯.𝑑𝑥 ﷯ ∴ ﷮﷮𝑥 log﷮𝑥﷯﷯﷮2﷯.𝑑𝑥﷯= ﷮﷮ log﷮𝑥﷯﷯﷮2﷯ 𝑥 .𝑑𝑥﷯ = log﷮𝑥﷯﷯﷮2﷯ ﷮﷮𝑥 .﷯𝑑𝑥− ﷮﷮ 𝑑 log﷮𝑥﷯﷯﷮2﷯﷮𝑑𝑥﷯ ﷮﷮𝑥 .𝑑𝑥﷯﷯﷯𝑑𝑥 = log﷮𝑥﷯﷯﷮2﷯ . 𝑥﷮2﷯﷮2﷯− ﷮﷮ 2 log﷮𝑥﷯﷯ 1﷮𝑥﷯ ﷮﷮𝑥 .𝑑𝑥﷯﷯﷯𝑑𝑥 = 𝑥﷮2﷯﷮2﷯ log﷮𝑥﷯﷯﷮2﷯−2 ﷮﷮ log﷮𝑥﷯﷮𝑥﷯ . 𝑥﷮2﷯﷮2﷯﷯ 𝑑𝑥 = 𝑥﷮2﷯﷮2﷯ log﷮𝑥﷯﷯﷮2﷯− ﷮﷮𝑥 log﷮𝑥﷯﷯ 𝑑𝑥 Now taking I1 = ﷮﷮𝑥 log﷮𝑥﷯﷯ 𝑑𝑥 ﷮﷮𝑥 log﷮𝑥﷯﷯ 𝑑𝑥= ﷮﷮ log﷮𝑥﷯﷯𝑥﷯𝑑𝑥 = log﷮𝑥﷯ ﷮﷮𝑥﷯𝑑𝑥− ﷮﷮ 𝑑 log﷮𝑥﷯﷯﷮𝑑𝑥﷯ ﷮﷮𝑥.𝑑𝑥﷯﷯𝑑𝑥﷯ = log﷮𝑥﷯ 𝑥﷮2﷯﷮2﷯﷯− ﷮﷮ 1﷮𝑥﷯ . 𝑥﷮2﷯﷮2﷯. 𝑑𝑥﷯ = 𝑥﷮2﷯﷮2﷯ log﷮ 𝑥﷯− 1﷮2﷯ ﷮﷮𝑥. 𝑑𝑥﷯ = 𝑥﷮2﷯﷮2﷯ log﷮𝑥﷯− 1﷮2﷯ . 𝑥﷮2﷯﷮2﷯ +𝐶 = 𝑥﷮2﷯﷮2﷯ 𝑙𝑜𝑔﷮ 𝑥﷯− 𝑥﷮2﷯﷮4﷯ +𝐶 Putting the value of I1 in (1), we get ﷮﷮𝑥 log﷮𝑥﷯﷯﷮2﷯.𝑑𝑥﷯= 𝑥﷮2﷯﷮2﷯ log﷮𝑥﷯﷯﷮2﷯− ﷮﷮ 𝑥 . log﷮𝑥﷯ 𝑑𝑥﷯ = 𝑥﷮2﷯﷮2﷯ log﷮𝑥﷯﷯﷮2﷯− 𝑥﷮2﷯ log﷮𝑥﷯﷯﷮2﷯ − 𝑥﷮2﷯﷮4﷯ +𝐶1﷯ = 𝑥﷮2﷯﷮2﷯ log﷮𝑥﷯﷯﷮2﷯− 𝑥﷮2﷯ log﷮𝑥﷯﷯﷮2﷯ + 𝑥﷮2﷯﷮4﷯ −𝐶1 = 𝒙﷮𝟐﷯﷮𝟐﷯ 𝒍𝒐𝒈﷮𝒙﷯﷯﷮𝟐﷯− 𝒙﷮𝟐﷯ 𝒍𝒐𝒈﷮𝒙﷯﷯﷮𝟐﷯ + 𝒙﷮𝟐﷯﷮𝟒﷯+𝑪

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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